Image data clustering and color conversion

ABSTRACT

Methods and systems herein provide for color conversion of image data to another color space. Such color conversion includes converting image data to a color space by identifying color regions in the image data. One color conversion system herein includes a color region identifier operable to select color values in the image data, determine numerical centers of the selected color values, and generate color regions, or color value “clusters”, based on the selected color values and the numerical centers. The color conversion system also includes an optimization module operable to refine the numerical center of each color region, determine a boundary for each color region based on the refined center, and remove a portion of the color cluster centers to optimize a number of the color regions for color conversion processing.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention generally relates to the field of printing systems, and in particular, to methods and systems for clustering image data and performing color conversion of color regions formed by the clustering.

2. Statement of the Problem

In color printing, displaying, and reproduction, the term gamut represents the set of colors that a color-reproduction device is physically able to generate. Every device that displays or reproduces an image, such as a printer, monitor, scanner, or digital camera, may have its own unique color gamut. When an image is transferred from one device to another, the color gamut of each device is examined to closely match the color in the transferred image. That is, an attempt is made to closely match the color gamut of the image originating device in the device to which the image is being transferred so as to provide the most aesthetically pleasing color conversion. For example, the color gamut of a digital camera is generally greater than the color gamut of a printer. When color values of the digital camera color gamut are mapped to the color gamut of the printer, the conversion process generally requires intense analysis to ensure that the print quality is sufficiently high.

Adding to the complexity are the color spaces themselves. For example, perceptual color spaces, such as CIELab, are visualized as three dimensional color spaces, where every color that humans can see is uniquely located. Though the CIELab color space is a perceptual color space, it is not a perceptually uniform color space as the Euclidean distance in the space does not correspond to the perceptual distance. For example, the magnitude of the perceptual color difference generally depends upon the color location and the changing direction in chroma and hue. CIELab increasingly overstates the magnitudes of perceived chroma differences. The human visual system (HVS) is sensitive to the change of chroma in the neutral color area and insensitive to the change of the chroma in a highly saturated color area. The CIELab color space is also non-uniform regarding hue angle in that the thresholds of visual tolerances are a function of hue angle. If the non-uniformity of the CIELab color space in chroma and hue is examined from another perspective, CIELab colors have different characteristics at different locations. The change of these characteristics is generally continuous.

Most printers (i.e., toner and ink based printers) are able to produce only a limited numbers of gray levels. In order to produce continuous tone imagery that contains an infinite range of colors or grays, a reprographic technique called halftoning is applied to create the illusion of continuous tone images through the use of dot arrangements and dots of varying size. The combination of the printer halftone design and the specific toner/ink selection determine the number of colors that a printer is physically able to produce (i.e., the gamut).

Printer color conversions are generally performed between a device-dependent color space and a device-independent color space. For example, CMYK color patches whose values span the CMYK color space are first printed. The printed patches are then measured using a spectral photometer which determines the spectral reflectance of each patch under a standard illuminant. Software in the spectral photometer calculates the tristimulus values and converts these values to CIELab values. The CIELab values are floating point numbers. Color conversion lookup tables are generated using this set of data to convert color values of the CIELab space to the CMYK space. However, the relationship between CMYK color space and the CIELab color space is highly nonlinear due to the interactions of cyan, magenta, yellow, and black planes. The color conversions, therefore, generally require complex functions. Another complication to the printer color conversion regards the halftone design. Most halftone techniques are capable of creating color gray levels continuously but have an abrupt change between two adjacent levels. Thus, the color conversion between CMYK and CIELab may be continuous but not differentiable.

Moreover, certain CMYK values may be mapped to the same CIELab value. Since the range of CIELab color space is generally much larger than the range of the CMYK color space, there are no CMYK conversions for many CIELab values. Additionally, the CIELab to CMYK conversion has one degree of freedom such that one CIELab value may be mapped to several CMYK values. Radial basis functions (RBFs) are useful in such a color conversion process because they are capable of approximating an unknown function from data. Additionally, RBFs generally do not require data to lie on any sort of regular grid.

Traditionally, weight parameters of the RBFs were determined by minimizing the mean-square error. Because the weight parameters may be determined using linear models, the RBFs generally require the use of clustering algorithms. However, due to the unique error characteristics in color conversion, the clustering algorithms require careful selection. Accordingly, there exists a need in selecting training data to identify clusters of color values in image data such that RBFs may be used for color conversion.

SUMMARY OF THE SOLUTION

Embodiments of the invention solve the above and other related problems with methods and associated systems by providing a color conversion system capable of converting image data to a color space by identifying color regions in the image data. One color conversion system herein includes a color conversion system operable to identify color regions in image data and convert the image data to a color space. The system includes a color region identifier operable to select color values in the image data, determine numerical centers of the selected color values, and generate color regions, or “clusters”, based on the selected color values and the numerical centers. The system also includes an optimization module operable to refine the numerical center of each color region, determine a boundary for each color region based on the refined center, and remove a portion of the numerical color region centers to optimize a number of the color regions for color conversion processing. For example, the optimization module may compute a covariance matrix of a color region and, based on a rank of that covariance matrix, remove the color region from subsequent processing to optimize the number of the color regions. The optimization module may configure the color values of the removed color region with an adjacent color region. The system may also include a converter operable to generate a numerical model of the color region using a radial basis function and convert the color region to a color space of an imaging device.

In another embodiment, a method of determining color regions in image data includes selecting clusters of color values in the image data, determining random center values of the clusters, iteratively determining boundaries of the clusters to define the color regions based on human perception properties, and processing subset color values of the clusters to remove clusters from subsequent processing.

In another embodiment, a printing system includes a printer operable to print to a tangible medium and a printer controller operable to process a print job from a host system and transfer the processed print job to the printer. The print job includes image data having color values. The printer controller includes a color region identifier operable to select the color values in the image data, determine numerical centers of the selected color values, and generate color regions based on the selected color values and the numerical centers. The printer controller also includes an optimization module operable to refine the center of the color region and determine a boundary for each color region based on the refined center and remove a portion of the numerical centers to optimize a number of the color regions for color conversion processing. The printer controller also includes a converter operable to convert the color regions to a color space prior to printing.

In another embodiment, a color conversion system operable to identify color regions in image data and convert the image data to a color space. The system includes a color region identifier operable to select clusters of color values in the image data and determine random center values of the clusters. The system also includes an optimization module operable to iteratively determine boundaries of the clusters to define the color regions and process subset color values of the clusters to remove clusters from subsequent processing.

The invention may include other exemplary embodiments described below.

DESCRIPTION OF THE DRAWINGS

The same reference number represents the same element or the same type of element on all drawings.

FIG. 1 is a block diagram of a color conversion system in one exemplary embodiment of the invention.

FIG. 2 illustrates a printing system employing a color conversion system in one exemplary embodiment of the invention.

FIG. 3 illustrates the non uniformity of CIELab chroma.

FIG. 4 is a graphical illustration of the Minimax Characterization Theorem (MCT).

FIG. 5 is a graph of exemplary toner/ink values used in color conversion in one exemplary embodiment of the invention.

FIG. 6 is a flowchart illustrating a method of converting color values of image data to a color space in one exemplary embodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

FIGS. 1-6 and the following description depict specific exemplary embodiments of the present invention to teach those skilled in the art how to make and use the invention. For the purpose of this teaching, some conventional aspects of the invention have been simplified or omitted. Those skilled in the art will appreciate variations from these embodiments that fall within the scope of the present invention. Those skilled in the art will appreciate that the features described below can be combined in various ways to form multiple variations of the present invention. As a result, the invention is not limited to the specific embodiments described below, but only by the claims and their equivalents.

FIG. 1 is a block diagram of a color conversion system 100 in one exemplary embodiment of the invention. In this embodiment, the color conversion system 100 includes a converter 103 that is used to convert the color values in image data to a desired color space using radial basis functions (RBFs). Generally, when using RBFs to solve color conversion problems, the color values in the image data require clustering. Examples of clustering algorithms that may be used in conjunction with the RBFs include the Voronoi based clustering algorithms and subset selection algorithms, such as an orthogonal least squares (OLS) algorithm. The Voronoi-based clustering algorithm is not optimized for the number of clusters in the image data and the subset selection algorithms generally depend on function selection and the norm selection in RBFs. To overcome this shortcoming, the system 100 may combine a Voronoi based clustering algorithm with a novel subset selection algorithm that is covariance-based so as to optimize the center selection and the number of clusters in the image data. The system 100 does so in a manner that is relatively independent of function and norm selection.

To implement the clustering and subset selection, the color conversion system includes a color region identifier 101 that is operable to select colors in the image data and determine a numerical center of the selected color values. From there, the color region identifier 101 generates a color region based on the selected color values and the numerical center. In doing so, the color region identifier 101 may take human perceptual color tolerance into consideration. For example, the color region identifier 101 may perform clustering on the CIELab data such that the characteristics of the image data can be best described by each cluster. Because the CIELab data is random, the color region identifier 101 may select a random center of a color region when initially clustering color values in the image data. The color region identifier 101 may then use the Voronoi-based clustering algorithm to initially identify the color regions within the image data.

The color conversion system 100 also includes an optimization module 102 that is operable to refine the center of the color region and determine a boundary of the color region based on the refined center such that color conversion processing may be performed by the converter 103. For example, the optimization module 102 may calculate the covariance matrix of each cluster. If the rank of a cluster's covariance matrix is less than the number of input color values, the center of that cluster is deleted and the color values thereof are moved to nearby, or adjacent, clusters. The optimization module 102 may update the center of each cluster and refine each cluster using a subset selection algorithm (e.g., via a covariance matrix technique or an orthogonal least squares technique). Generally, the optimization module 102 repeats this process until every cluster has a full rank covariance matrix. In this regard, the optimization module 102 may also optimize the number of clusters. For example, the optimization module 102 may remove clusters from processing if the clusters do not accurately define a color region. The color values of the removed clusters may then be reallocated to adjacent clusters such that they may be processed in the remaining color regions. The parsed color regions may then be transferred to the converter 103 for RBF-based color conversion. The details of clustering, subset selection, and RBF-based color conversion are described in greater detail below.

FIG. 2 illustrates a printing system 200 employing a color conversion system 100 that is used to convert color values in image data to the color gamut of a printer, in one exemplary embodiment of the invention. In this regard, the color conversion system 100 may be implemented as software instructions that direct the printer controller 202 to perform the color conversion in a manner as described in the system 100 of FIG. 1. The printing system 200 receives print jobs from one or more host systems 201 that are communicatively coupled to the printing system 200. Thus, when a host system user wishes to print on the two-sided medium 213, the host system user may generate a print job that directs the rasterizer 208 to do so. The printing system 200 includes a printer controller 202 that receives the print job and rasterizes the print job via the rasterizer 208. Rasterization generally regards the conversion of an image described in a vector graphics format into a raster image of pixels such as a bitmap for output to a video display or printer. Examples of the two-sided medium 213 include paper, transparent film, and any other type of material capable of being imprinted.

Prior to rasterization, the system 100 receives the image data of the print job and performs a color conversion of the color values therein such that they are aesthetically presentable on the two-sided medium 213. As mentioned, the system 100 may direct the printer controller 202 to process image data having a plurality of color values and randomly select clusters of the color values in the image data to identify color regions in the image data. In doing so, the printer controller 202 may determine centers of the clusters and iteratively determine boundaries of the clusters to define the color regions (e.g., using a Voronoi set of the selected color values and refining the center values via an OLS algorithm). The printer controller 202 may also optimize the number of clusters by processing subset color values of the clusters to remove certain color regions from subsequent processing. For example, the printer controller 202 may optimize the identified color regions by generating a covariance matrix of each cluster, determining a rank of each covariance matrix, and analyzing the rank of each covariance matrix to determine subsequent processing of the respective cluster. In response, the printer controller 202 may remove certain clusters from subsequent processing and move color values of the removed cluster(s) to adjacent clusters. The printer controller 202 may repeat this process until each cluster includes a full rank covariance matrix. Once the printer controller 202 identifies the color regions and optimizes these regions, the printer controller 202 may convert the color values therein to a desired color space via RBFs as generated by the converter 103.

Those skilled in the art should readily recognize that the printer controller 201 may be implemented in software, hardware, firmware, or a combination thereof to perform the various functions associated with the printer controller 202 including the novel aspects of color conversion as described herein.

Before discussing the numerical models for CIELab to CMYK and CMYK to CIELab color conversions in detail, a discussion of color spaces is now presented to guide the reader through an understanding of color conversions and gamut mapping. Perceptual color spaces, such as CIEXYZ and CIELab, are visualized as three dimensional color spaces, where every color that humans can see is uniquely located. The location of any color in the space is determined by its color coordinates. In the CIEXYZ color space, the components are X, Y, and Z, where Y is luminance, X and Z do not correlate to color appearances. In the CIELab color space: the components are L*, a*, and b*, where L is lightness of the color (e, g., L*=0 yields black and L*=100 indicates white), a* is the value between red and green (e.g., negative values indicate green and positive values indicate red), and b* is the value between yellow and blue (e.g., negative values indicate blue and positive values indicate yellow). The CIELab color space can also be described using cylindrical coordinates in the L*C*h color space where C*_(ab) is the chroma coordinate, and h is the hue coordinate. Below are the mathematical expressions to convert a* and b* into C*_(ab) and h:

$\begin{matrix} {C_{ab}^{*} = \sqrt{a^{*2} + b^{*2}}} & \left( {{Eq}.\mspace{14mu} 1} \right) \\ {h = {\tan \left( \frac{b^{*}}{a^{*}} \right)}} & \left( {{Eq}.\mspace{14mu} 2} \right) \end{matrix}$

The relationship between CIEXYZ and CIELab may be described using the following equations:

$\begin{matrix} {{L^{*} = {{116{f\left( \frac{Y}{Y_{n}} \right)}} - 16}},} & \left( {{Eq}.\mspace{14mu} 3} \right) \\ {{a^{*} = {500\left( {{f\left( \frac{X}{X_{n}} \right)} - {f\left( \frac{Y}{Y_{n}} \right)}} \right)}},} & \left( {{Eq}.\mspace{14mu} 4} \right) \\ {{b^{*} = {200\left( {{f\left( \frac{Y}{Y_{n}} \right)} - {f\left( \frac{Z}{Z_{n}} \right)}} \right)}},} & \left( {{Eq}.\mspace{14mu} 5} \right) \end{matrix}$

where X_(n), Y_(n), Z_(n) are the tristimulus values of a reference white and generally referred to as the white point, described as

$\begin{matrix} {{f(x)} = \left\{ \begin{matrix} x^{\frac{1}{3}} & {x > 0.008856} \\ {{{7.787x} + \frac{16}{116}},} & {x \leq {0.008856.}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 6} \right) \end{matrix}$

The color difference in either CIEXYZ or CIELab is generally calculated using the Euclidean distance. Though the CIEXYZ and CIELab color space are perceptual color spaces, they are not perceptually uniform color spaces as the Euclidean distance in the space does not correspond to perceptual distance. For example, the colors that have the same Euclidean distance with respect to an anchor color are on a circle. However, these circles are transformed to ellipsoids when the human perceptual distances are measured. The major axis of the ellipsoid represents the higher tolerance of the color difference and the minor axis of the ellipsoid represents the lower tolerance for the color difference. The CIELab color space is a perceptually more uniform color space than the CIEXYZ color space. Thus, the CIELab color space is the color space used most often for color-difference evaluations.

In the CIELab color space, the magnitude of the perceptual color difference generally depends upon the color location and the changing direction in chroma and hue. CIELab increasingly overstates the magnitudes of perceived chroma differences. For example, the human visual system (HVS) is sensitive to the change of chroma in the neutral color area. The HVS is insensitive to the change of the chroma in a highly saturated color area. FIG. 3 illustrates such change with the ellipsoid size increases 301.

The CIELab color space is also non-uniform regarding hue angle in that the thresholds of visual tolerances are a function of hue angle. If the non-uniformity of the CIELab color space in chroma and hue is examined from another perspective, CIELab colors have different characteristics at different locations. The change of these characteristics is generally continuous. Color difference models correlate the perceptual distance with the Euclidean distance in the CIElab color space based on vast amounts of color difference experimental data.

There are several CIE perceptual-based color difference models. For example, CIEDE2000 calculates the perceptual difference by weighted lightness difference, chroma difference, hue difference, and the interaction between the chroma and the hue difference. However, the CIEDE2000 was developed for small color differences and does not perform well for large color differences. The empirical weighted color differences of lightness, chroma, and hue are often used for CIEDE2000, with higher weighting factors for hue and lightness differences and lower weighting factor for chroma differences. The color difference calculated using the Euclidean distance in CIELab is called deltaE, (also referred as ΔE, or dE_(ab).) dE₀₀ represents the color difference calculated using CIEDE2000.

In general, there are two types of color management systems among color devices: additive systems such as a computer monitor and subtractive systems such as a printer. In additive systems, colors are transmitted lights. Red, green and blue lights are referred as the additive primary colors. When used in various degrees of intensity and variation, they create all other colors of light. When superimposed equally, they create gray. Primary colors of subtractive systems like printers may be cyan, magenta, yellow, and black that are used together to effectively create a multitude of other colors based on the subtractive color theory.

The color conversions for an additive system are normally between the RGB color space and CIEXYZ color space. These conversions are relatively straightforward since the gamma corrected RGB primaries are linearly related to XYZ values in the CIEXYZ space for most devices. An example of such conversion is the conversion for the RGB colors described as follows:

$\begin{matrix} {{\begin{pmatrix} X \\ Y \\ Z \end{pmatrix} = {\begin{pmatrix} 0.4124 & 0.3576 & 0.1805 \\ 0.2126 & 0.7152 & 0.0722 \\ 0.0193 & 0.1192 & 0.9505 \end{pmatrix}\begin{pmatrix} {g\left( R_{srgb} \right)} \\ {g\left( G_{srgb} \right)} \\ {g\left( B_{srgb} \right)} \end{pmatrix}}},{where}} & \left( {{Eq}.\mspace{14mu} 7} \right) \\ {{g(k)} = \left\{ {{{\begin{matrix} {\left( \frac{K + \alpha}{1 + \alpha} \right)^{\gamma},} & {K > 0.4045} \\ {\frac{K}{12.92},} & {{otherwise},} \end{matrix}\gamma} = 2.2},{{{and}\mspace{14mu} \alpha} = {0.055.}}} \right.} & \left( {{Eq}.\mspace{14mu} 8} \right) \end{matrix}$

As is known, color printers may be toner-based or ink-based. Most of the printers are only able to produce a limited numbers of gray levels (e.g., a binary printer produces two levels (i.e., 0 and 1) 2-bit printers produce levels of 0 1, 2, and 3). In order to produce continuous tone imagery that contains an infinite range of colors or grays, a reprographic technique called halftoning is applied to create the illusion of continuous tone images through the use of dot arrangements and dots of varying size. The combination of the printer halftone design and the specific toner/ink selection determine the number of colors that a printer is physically able to produce which is called its gamut. The gamut is measured in the CIELab color space.

Printer color conversions depend the toner/ink selections and the halftone design. The conversions are generally performed between a device-dependent color space (CMYK) and a device-independent color space (CIELab). Often the format of the color conversion is an equally spaced lookup table, the values of which are either 1-byte or 2-byte integers.

In generating a printer color conversion, CMYK color patches whose values span the CMYK color spaces are printed. Each component value of CMYK is normally a 1-byte integer. The printed patches are measured using a spectral photometer which measures the spectral reflectance of each patch under a standard illuminant. The software in the spectral photometer calculates the tristimulus values and converts these values to CIELab values. The CIELab values are floating point numbers. Followed by the implementation of a data modeling algorithm and a gamut mapping algorithm, the color conversions LUTs are generated using this set of data. The CMYK and the CIELab values are normalized to the range of 0 to 1 for these calculations. Both CMYK to CIELab and CIELab to CMYK conversions may be generated for a color printer. The CMYK to CIELab color conversion is used when an image generated in this CMYK color space is to display on a monitor or to print on a different color printer. In other words, this conversion is used when the printer CMYK color space is used as a source color space. The CIELab to CMYK color conversion is used when the CIELab color space is the source color space.

For many color printers, the relationship between CMYK color space and the CIELab color space is highly nonlinear due to the interactions of cyan, magenta, yellow, and black planes in the printer subtractive system, thereby requiring complex functions. Another complication to the printer color conversion is due to the halftone design. Most halftone techniques are capable of creating color gray levels continuously. However, an abrupt change of two adjacent levels is a weakness in many halftone designs. Thus, the color conversion between CMYK and CIELab may be continuous but not differentiable.

The CMYK to CIELab color conversion is neither injective nor subjective. For example, every color in the CMYK color space may be mapped to a color in the CIELab color space. However, different CMYK values may also be mapped to the same CIELab value. So the CMYK to CIELab color conversion is not injective. The range of CIELab color space is much larger than the range of CMYK color space. That is, for many CIELab values, there are no CMYK values such that f(C, M, Y, K)→(L, a, b). Thus the CMYK to CIELab color conversion is not subjective.

Additionally, the CIELab to CMYK conversion is from R3 to R4. Thus, only one degree of freedom in this conversion exists and one CIELab value may be mapped to several CMYK values. On the other hand, because of the smaller gamut of the color printer, not every CIELab value can be mapped to a CMYK value. Since no function exists for the CIELab to CMYK color conversion, the conversion is achieved by converting the CIELab values to CMY values first and a special rule is created for adding the black toner to each CMY value.

The CIELab to CMY color conversion within a printer gamut is both injective and subjective. That is, virtually every CIELab value inside the printer color gamut can be mapped to a unique color in the CMY color space. So the CIELab to CMY color conversion is injective within color gamut. Additionally, for CMY values, there exists a CIELab value such that g (L, a, b) (C, M, Y) where the CIELab to CMY color conversion is subjective. Out-of gamut CIELab values generally cannot be mapped to a CMY value. Therefore, no functions exist for the CIELab to CMY conversion outside the printer gamut and color conversion between the CMYK and CIELab color spaces is achieved by approximation.

Understanding the sources of error is important for the design of an approximation model. In addition, it is important to understand how these errors interfere with human perceptions for the design of the printer color conversion model. There are four significant sources of error in the color conversion process. These errors are normally measured with the Euclidean distance in the CIELab color space, ΔE, and include:

-   -   1. measurement errors from the instrument, which are normally         smaller than 1 ΔE;     -   2. errors due to the machine reliability and repeatability,         which for a well-calibrated printer are generally smaller than 2         units of ΔE;     -   3. errors due to paper roughness and non-uniform paper fibers         and fillers, which are typically smaller than 1 unit of ΔE; and     -   4. errors from the model of the color conversion, which average         around 3-4 units of ΔE with a maximum error between about 10 to         20 units of ΔE.

Because human perceptual tolerance varies from location to location in the CIELab color space, care is taken in the approximation algorithms in different color areas. For colors in neutral areas, the visual color difference tolerance is low (around 1 ΔE). The interpolation errors in these regions are generally small. If the accumulated errors from the first three categories are added to the modeling data, the effect of the outliers in the modeling set should be minimized. For the medium colors, with the increase of the chroma values in these areas, the perceptual tolerance increases. Though the outliers still exist, they play a much less important role in the color conversion. It is appropriate to design an approximation model to minimize the least squares error. For colors in the highly saturated color regions, interpolation errors are relatively high. The approximation algorithm is used to minimize the maximum error in the interpolation.

In general, color conversion approximation algorithms aim at minimizing least square errors. Due to the sources of error described above, emphasis is placed on the L_(p) error criteria, in particular the L₁ and L_(∞) error for a robust neutral area color conversion and the reduction of maximum error in the saturated color regions. One approximation model includes the natural neighbor interpolation method, which is based on the Voronoi region. A Voronoi region V_(i) is a set of points for which a center c_(i) is the nearest center vector, as follows:

V _(i) ={xεR ^(n) :i=arg min_(jεI) ∥x−c _(j)∥},  (Eq. 9)

where I is the set of center indices. This interpolation method is a local region-based weighting method generally provides better underlying printer characteristics.

Global interpolation methods may also include the inverse distance weighting method and the RBF method. For the inverse distance weighting method, data points are used to interpolate a value by assuming that interpolated values are affected more by nearby points and less by more distant points. The assumption is not accurate for colors in many regions. Thus, this method has yielded relatively poor color conversion results. RBF data fitting equipped with an L_(p) error-criterion, on the other hand, improves color conversion. The embodiments described herein also take into account perceptual color difference tolerance. As mentioned, perceptual color difference tolerances generally vary over different CIELab color regions. Taking this into account, the RBF data fitting technique has the advantage of minimizing color differences in different areas with respect to human perceptions or yield results.

Although black toner/ink is one of the primaries in a color printer, the black toner/ink actually serves as a “secondary” toner in color printing. Because the mix of cyan, magenta, and yellow toner/ink produces a “muddy brown” instead of black, black toner/ink is added in to make a more rich black and thereby make other colors richer and darker. Black toner/ink also typically costs less than cyan, magenta, and yellow toner/ink making the embodiments herein more advantageous from a cost perspective.

Printer CIELab to CMYK color conversion maps from R3 to R4. Additionally, CIELab is mapped to the CMY color space using an interpolation technique, then black toner/ink denoted by K, is added. The amount of cyan, magenta, yellow toner/ink may be adjusted by applying the under color removal (UCR) and the under color addition (UCA) corrections. The typical way of applying UCR is to remove equal amounts of yellow, magenta, and cyan and replace them with the same amount of black. Gray component replacement (GCR) is the result of UCR followed by UCA. However, colors resulting from UCR are generally dark, less saturated, and hue shifted. UCA may be applied to improve the weakness of UCR. Small portions of the three colors of cyan, magenta, and yellow may be added back to these areas. The UCA process is empirical and relatively processor intensive due to the mapping from a lower dimensional space to a higher dimensional space (e.g., R3 to R4) and the lack of a high fidelity model describing the nonlinear behavior of each toner. UCA is normally achieved by printing many patches and choosing the best results. In practice, the UCR and UCA correction schemes introduce errors in the color conversion process.

Because the CIELab to CMYK conversion is a map from R3 to R4, a set of CMYK values corresponding to the same CIELab value may be established. In this regard, g denotes a mapping from the CIELab values to the CMYK values such that

∃{(C,M,Y,K)^((i)) ,i=1,2, . . . k},s.t.,g ⁻¹(C,M,Y,K)^((i))=(L,a,b),  (Eq. 10)

where, i=1; 2; . . . ; k. These level sets of CMYK values may be found through an optimization process if approximation models are developed for the CMYK to CIELab conversion and the CIELab to CMY conversion.

As mentioned, gamut mismatch is problematic in color reproduction systems since each device has a different gamut. In general input devices such as digital cameras, scanners, etc., generate more colors than the output devices such as color printers. When the input color space is larger than the gamut of the output color device, gamut mapping algorithms are applied. The gamut mapping process transforms a point in the source gamut to a realizable color inside the gamut of the output device (e.g., a destination gamut). The form of this transformation can impact the quality of the reproduced images, especially print images. For high-end digital printers, the gamut is relatively small. The maximum error of color conversions normally derives from mapping an outside-gamut color to an inside-gamut color mapping.

In the gamut mapping process, a device color gamut boundary may be initially identified. The calculated boundaries are normally connected lines. Previous gamut mapping algorithms mapped out-of-gamut points directly to the destination gamut. The most typical gamut mappings include clipping and scaling algorithms. Clipping algorithms clip out-of-gamut points to the destination gamut boundary. Scaling algorithms, on the other hand, scale the input color gamut to output color gamut such that some out-of-gamut points are mapped inside the destination gamut while others are mapped to the boundary of the destination gamut. The mapping directions may include a choice of lines along which the mapping is applied (e.g., a line changing in chroma at constant lightness and hue) that are often decided based on visual experiments. The mapping directions may also be along the lines of constant perceptual attribute predictors (e.g., constant lightness and hue, constant saturation and hue), along the lines towards a single center-of-gravity (e.g., the center is the compression towards L*=50 on lightness axis), along the lines towards variable center-of-gravity (e.g., the cusp is the compression towards lightness of cusp on lightness axis), or along the lines towards the nearest color in the reproduction gamut (e.g. as is the case with minimum ΔE clipping).

The large variability in past color gamut mapping suggests that no universal gamut mapping algorithm works for every device and image. Ideal gamut mapping generally depends on image content, preservation of perceived hue throughout color space, and the extent of the gamut mismatch in various regions of color space. Relationships among color data developed by a numerical model have not previously been considered in the gamut mapping process. The underlying relationship among color data may more accurately retain or rationally transform after gamut mapping. Such is achieved herein using a numerical model with guidance from the perceptual model.

The numerical model quantitatively compares the color difference between the original color and the printed color. The original colors are the CIELab colors. These colors are converted with the printer color conversion to the CMYK colors. The CMYK colors are sent to the printer to print and then measured by spectral photometers. The numerical method tests the approximation errors inside color gamut and gamut mapping results out-side the color gamut. The mean error and the maximum error may be calculated using either dE_(ab) or dE₀₀ for the color conversion inside the printer gamut. The maximum conversion error is often weighted more than the mean error in the evaluation because the color difference is more objectionable than the average color difference.

The perceptual model, on the other hand, correlates human perceptions with physical stimuli. A set of test images including pictorial images, vector graphics, and text samples are printed from the printer. The colors in the images include highlight colors, low-light colors, saturated colors, and out-of-gamut colors. Observers look at these images and make perceptual judgments based on the selected psychophysics method, such as the paired comparison method. On such experiments, observers are asked to compare an image pair converted by different color conversion methods. The original photos are normally presented as the reference. Typical questions are “Which one do you like better?”, “Which one is more similar to the original image?”, etc. The data is analyzed and converted to perceptual values using Thurstone's law of comparative judgment.

To assist the reader in the development of the numerical model, this disclosure broken into certain sections. Specifically, the sections are entitled “FITTING NON-LINEAR SCATTERED DATA”, “OVERDETERMINED LINEAR SYSTEMS IN THE L_(∞) SENSE”, “COLOR CONVERSION USING THE L₁ NORM APPROXIMATION”, “ONE DIMENSIONAL COLOR MANIFOLDS IN THE CMYK COLOR SPACE”, “COLOR CONVERSION WITH INK LIMITATIONS”, and “GAMUT MAPPING”. The section entitled “FITTING NON-LINEAR SCATTERED DATA” provides the reader with a detailed description of solving printer color conversion problems using RBFs and data fitting techniques adapted to the L_(∞) and the L₁ error criteria. The section entitled “OVERDETERMINED LINEAR SYSTEMS IN THE L_(∞) SENSE” presents the reader with the Barrodale and Phillips dual algorithm (the “BP” algorithm) and the Bartels, Conn, and Li primal algorithm (the “BC” algorithm) for converting relatively large sparse problems to linear programming problems. The section entitled “COLOR CONVERSION USING THE L₁ NORM APPROXIMATION” illustrates a general L₁ approximation for overdetermined linear systems, the BR algorithm for solving the primal problem, and the L₁ approximation as it is used in color conversion. The section entitled “ONE DIMENSIONAL COLOR MANIFOLDS IN THE CMYK COLOR SPACE” introduces the reader to an optimized GCR algorithm with relatively high quality toner/ink selection. The section entitled “COLOR CONVERSION WITH INK LIMITATIONS” discusses toner/ink limitations implemented as a constraint in the linear programming problem formulations. The section entitled “GAMUT MAPPING” introduces the reader to the concept of numerical gamut mapping based on the RBFs discussed above and enhanced by the perceptual gamut mapping.

Fitting Non-Linear Scattered Data

Problems with the interpolation techniques are overcome through the use of RBFs that include data fitting techniques adapted to L_(∞) and L₁ error criteria. Additionally, a perceptual color model is used to guide the implementation of the RBFs. For example, in order to solve nonlinear approximation problem using RBFs, a clustering algorithm is employed using the combination of the Linde-Buzo-Gray (LBG) algorithm and the covariance matrix-based algorithm. RBFs may then be selected and parameters thereof may be optimized. Approximation algorithms are generated for the L_(p) norm (emphasis on the L_(∞), L₂, and L₁ norm) for the neutral, intermediate, and saturated color areas. A K-fold cross validation then refines the model parameters and an optimized overlapping area between two different models is determined.

Theoretically, a color printer is capable of producing at least 4 bytes of colors (1 byte each of cyan, magenta, yellow, and black). A subset of these CMYK colors and their associated CIELab values may be used to generate a color conversion model. Because of the underlying complicated characteristics of a color printer, a large CMYK data set and the associated CIELab values (often above 2000 data points) are selected to generate the color conversion model. The data selection depends in part on the halftone design which occasionally yields a nonlinear response. The data selection also depends on the perceptual color difference tolerance as the perceptual color difference tolerances vary over different CIELab color regions. Therefore, different color conversion models should be generated for different regions based on the perceptual color difference tolerance. The continuity and smoothness of these color conversion models are important to the overall quality of the color conversion. RBFs provide the flexibility of applying to both the interpolation and approximation data fitting problems. RBFs are useful to interpolate scattered data as the associated system of linear equations is invertible under relatively mild conditions on the locations of the data points.

Generally, RBFs are useful for approximating an unknown function from the data in the form

$\begin{matrix} {{{f(x)} = {{Ax} + a_{0} + {\sum\limits_{i = 1}^{N}{w_{i}{\varphi \left( {{x - c_{i}}} \right)}}}}},} & \left( {{Eq}.\mspace{14mu} 11} \right) \end{matrix}$

where f(x) is assumed to be continuous and differentiable over a compact domain of xεR^(n) and ƒ(x)εR. The term Ax represents the linear component of the mapping and the offset a₀ allows for the mean of the data to be non-zero. The nonlinear portion of the map is represented by the superposition of the nonlinear function φ weighted by vectors w_(i).

The RBF has several important features. For example, the weight parameters may be determined using linear models, the RBFs may be either local or global and the locations of the basis functions may be adapted using different clustering routines. For color conversion, the Voronoi region, for example, may provide relatively good clustering because it can incorporate the color difference model to characterize each cluster with its unique characteristics. Also, RBFs generally do not require data to lie on any sort of regular grid and the function ƒ(x) is continuous and differentiable.

Traditionally, the weights w_(i) were determined by minimizing the mean-square error. However, due to the unique error characteristics in color conversion, the algorithms for solving the RBFs' weights using L_(p) norm emphasizes L_(∞) norm, L₂ norm, and L₁ norm. Accordingly, the approach described herein includes choosing a training/testing data set for color conversion, clustering the color data, determining the centers optimizing the number of centers, and optimizing RBFs and parameter selections. Then, the L_(p) norm selections (specifically, the L_(∞), L₂, and L₁ norms) may be optimized and the RBFs combined with linear programming for the L_(∞) norm and L₁ norm approximation employing the duality theorem. Thereafter, the perceptual color model may be employed to improve the qualitative result.

The color conversion model is built based on the availability of a data set of input-output pairs, such as CMYK vs. CIELab. There are many ways to select a set of training data. For example, the CMYK colors in regular grids may be selected or the CMYK colors based on the halftone curves may be selected. In any case, the CMYK data is printed on a color printer and measured in the CIELab color space. For the CMYK to CIELab conversion, these CIELab values are random within the printer gamut and can be expressed for f: R4→R3 in the following form:

{x ^((i))=(C,M,Y,K)^((i)) εR4,i=1,2, . . . , m}  (Eq. 12)

{y ^((i))=(L,a,b)^((i)) εR3,i=1,2, . . . , m},  (Eq. 13)

where m is the number of color pairs in the data set. For the CIELab to CMYK conversion, g: R4^(→)R3

{x ^((i))=(L,a,b)^((i)) εR3,i=1,2, . . . , m}  (Eq. 14)

{y ^((i))=(C,M,Y,K)^((i)) εR4,i=1,2, . . . , m}.  (Eq. 15)

The printer color conversions described herein include both CMYK to CIELab and the CIELab to CMYK color conversions. The CIELab to CMYK conversion is achieved by converting the CIELab values to the CMY values first and then adding black to each CMY value via the GCR process. Therefore, the CMYK set should have enough C, M, Y, and K values. Generally, the grid points of cyan, magenta, yellow, and black components are used to represent the curve change of each component created by the halftone process. This means that the grids point may not be equally spaced.

Data clustering is used to identify the locations where basis functions should be placed in the model. In general, RBF algorithms focus on placing functions such that the mean square error is minimized. Due to the uniqueness of perceptual color characteristics, the location of the basis function should be placed in the local regions which have similar properties in terms of human visual tolerances. As mentioned, the CMYK values are device-specific values and do not directly relate to human perception. In the construction of the mapping g: R4^(→)R3 and ƒ: R4^(→)R3, clustering is performed on the CIELab data so the characteristics of the color data can be described by each cluster. The visual tolerances of color data may be determined using a color difference model. The most recent color difference model CIEDE2000 is used as the distance metric for each color relative to the center color because the distances are designed to be small.

The clustering algorithm also combines the Voronoi-based clustering algorithm with a subset selection algorithm. Because the CIELab data are random, a random center selection algorithm may be used as the first step for clustering/center selections. The clustering algorithms are then used to identify the clusters and to refine the center of each cluster. There are several efficient clustering algorithms available that may be used including “k-means,” LBG, and topology-preserving mappings. The algorithm generally proceeds by selecting an initial number N_(c) of centers randomly from the m data points in the training set and, for each center, computing the Voronoi set (i.e., all points in the data closest in the perceptual color space to this center relative to other centers). Then, the centers as the mean of the points in the Voronoi set are updated. This is repeated until the CIELab values of these centers no longer change. The clustering algorithm described above can be effectively combined with a subset selection algorithm, such as the orthogonal least squares (OLS) algorithm or the covariance matrix algorithm of commonly owned and co pending U.S. Patent Application Publication No. 2007/009796, the entire contents of which are incorporated by reference, to eliminate unnecessary clusters.

The OLS algorithm identifies which of the N centers are most useful in the RBF model and indicates which centers may be deleted from the model and at what expense to accuracy expressed in color difference. The computation of the weight parameters in Eq. 11 is an over-determined least squares problem. Thus, a solution is sought to the set of inconsistent equations,

y=Φw,  (Eq 16)

where each column of the matrix Φ is associated with a single center and ƒ does not actually reside in the column space of Φ. In general, a quality function is associated with a center. In this case, the value of a column ø_(i) of Φ is measured to solve the least squares. One measure is the cosine of the angle between ƒ and each column of Φ, such as

$\begin{matrix} {v_{i} = {\left( \frac{f^{T}\varphi^{(i)}}{{f}{\varphi^{(i)}}} \right)^{2}.}} & \left( {{Eq}.\mspace{14mu} 17} \right) \end{matrix}$

In other words, the center is the one for which v_(i) is largest (i.e., where the angle is smallest). This solution is indexed by i* meaning ø_(i)* is the solution to optimization. Once the column and hence the RBF center of Φ has been determined, the next best center can be computed by projecting the remaining columns along ø_(i)*. This approach may be iterated to obtain a reduced subset of centers that has acceptable modeling accuracy. This algorithm may therefore proceeds as follows:

Let k=1, 2, . . . , N_(c)

Step 1: k=1

$\begin{matrix} {i_{1} = {\arg \; {\max \left( \frac{f^{T}\varphi^{(i)}}{{f}{\varphi^{(i)}}} \right)}^{2}}} & \left( {{Eq}.\mspace{14mu} 18} \right) \\ {q^{(1)} = \varphi^{(i_{1})}} & \left( {{Eq}.\mspace{14mu} 19} \right) \end{matrix}$

Step k=1

$\begin{matrix} \begin{matrix} {{\alpha_{jk}^{(i)} = \frac{q_{j}^{T}\varphi_{i}}{q_{j}^{T}q_{j}}},} & {1 \leq j < k} \end{matrix} & \left( {{Eq}.\mspace{14mu} 20} \right) \\ {q_{k}^{(i)} = {\varphi_{i} - {\sum\limits_{j = i}^{k - 1}{\alpha_{jk}^{(i)}q_{j}}}}} & \left( {{Eq}.\mspace{14mu} 21} \right) \\ {i_{k} = {\arg \; {\max \left( \frac{f^{T}q_{k}^{(i)}}{{f}{q^{(i)}}} \right)}^{2}}} & \left( {{Eq}.\mspace{14mu} 22} \right) \end{matrix}$

where i_(k)≦i≦N_(c), i≠i₁, . . . i≠i_(k−1), and

$\begin{matrix} {q_{k} = {q_{k}^{i_{k}} = {\varphi_{i_{k}} - {\sum\limits_{j = 1}^{k - 1}{\alpha_{jk}{q_{k}.}}}}}} & \left( {{Eq}.\mspace{14mu} 23} \right) \end{matrix}$

The algorithm ends at the Mth step when

$\begin{matrix} {{{1 - {\sum\limits_{j = 1}^{M}\left( \left( \frac{f^{T}q_{j}}{{f}{q_{j}}} \right)^{2} \right)}} < \rho},{{{where}\mspace{14mu} 0} < \rho < 1}} & \left( {{Eq}.\mspace{14mu} 24} \right) \end{matrix}$

A cluster can also be described by its mean vector and variance-covariance matrix. The mean vector is the center of the cluster and the variance-covariance matrix consists of the variances of the color data points along the main diagonal and the covariance between each pair of color components in the other matrix positions. For example, if a cluster is in the CMYK color space, the covariance of this cluster provides a measure of the strength of the correlation between any two color components. Let the center CMYK value of a cluster be x= C, M, Y, K) (i.e., x ₁= C, x ₂= M, x ₃= Y, x ₄= K) and let m_(c) denote the number of data points in a cluster. The formula for computing each element COV_(jj′) in a 4×4 CMYK covariance matrix for a cluster is:

$\begin{matrix} {{{COV}_{{jj}^{\prime}} = {\sum\limits_{i = j}^{mc}{\left\lbrack {\left( {x_{j}^{i} - {\overset{\_}{x}}_{j}} \right)\left( {x_{j^{\prime}}^{i} - {\overset{\_}{x}}_{j^{\prime}}} \right)} \right\rbrack/\left( {m_{c} - 1} \right)}}},} & \left( {{Eq}.\mspace{14mu} 25} \right) \end{matrix}$

where j; j′=1, . . . , 4. When the rank of the covariance matrix is less than the number of the input color components, the correlations are not all independent of each other. Also, the number of the data points in the cluster is less than the number of color components.

A Gaussian function which can be expressed as the function of the covariance matrix is not selected if a non-full rank cluster exists. Thus, the clustering algorithm proceeds by calculating the covariance matrix of each cluster and, if the rank of the covariance matrix of a cluster is less than the number of input color components, the center of this cluster is deleted and the data points in the cluster are moved to nearby clusters. Thereafter, the center of each cluster is updated and the LBG algorithm is updated to refine each cluster. This is repeated until every cluster has a full rank covariance matrix.

The radial basis functions themselves are generally selected so that they satisfy an invertibility condition on the square interpolation problem for the weights. These functions have expansions, such as those given by Eq. 11, that represent continuous functions over compact domains. The location of these functions is stipulated by the vector centers (ck) generally scattered over the domain in a manner that reflects the distribution of the data. Both theoretical investigation and practical results suggest that the choice of the nonlinearity ø(.) is not critical to the performance of the RBF. However, for the color conversions from CMYK to CIELab, and CIELab to CMYK, the multiquadric function

φ(x)=√{square root over (r ² +x ²)}  (Eq. 26)

is generally more accurate. Other functions that may be used include the Gaussian

$\begin{matrix} {{\varphi (x)} = {\exp \left( \frac{- x^{2}}{r^{2}} \right)}} & \left( {{Eq}.\mspace{14mu} 27} \right) \end{matrix}$

and the inverse multiquadratic function

$\begin{matrix} {{\varphi (x)} = {\frac{1}{\sqrt{r^{2} + x^{2}}}.}} & \left( {{Eq}.\mspace{14mu} 28} \right) \end{matrix}$

Regardless of the function used, the radius r in Gaussian, multiquadratic, and inverse multiquadratic is optimized. Generally, the multidimensional Newton's method is an efficient optimization method for 2-norm least square settings. In this case, the objective function aims at minimizing the mean error. For example, using the color conversion from CMYK to CIELab:R4→R3,

$\begin{matrix} {{f_{L}\left( {x_{i},r_{L}} \right)} = {{A_{L}x_{i}} + a_{L\; 0} + {\sum\limits_{j = 1}^{n}{w_{Lj}{\varphi_{L}\left( {{{x_{i} - c_{j}}},r_{L}} \right)}}}}} & \left( {{Eq}.\mspace{14mu} 29} \right) \\ {{f_{a}\left( {x_{i},r_{a}} \right)} = {{A_{a}x_{i}} + a_{a\; 0} + {\sum\limits_{j = 1}^{n}{w_{aj}{\varphi_{a}\left( {{{x_{i} - c_{j}}},r_{a}} \right)}}}}} & \left( {{Eq}.\mspace{14mu} 30} \right) \\ {{f_{b}\left( {x_{i},r_{b}} \right)} = {{A_{L}x_{i}} + a_{b\; 0} + {\sum\limits_{j = 1}^{n}{w_{bj}{\varphi_{b}\left( {{{x_{i} - c_{j}}},r_{b}} \right)}}}}} & \left( {{Eq}.\mspace{14mu} 31} \right) \end{matrix}$

The problem formulation is thus:

$\begin{matrix} {{E\left( {r_{L},r_{a},r_{b}} \right)} = {\sum\limits_{i = 1}^{m}\begin{bmatrix} {\left. {{f_{L}\left( {x_{i},r_{L}} \right)} - y_{1}^{i}} \right)^{2} +} \\ {\left. {{f_{a}\left( {x_{i},r_{a}} \right)} - y_{2}^{i}} \right)^{2} +} \\ \left. {{f_{b}\left( {x_{i},r_{b}} \right)} - y_{3}^{i}} \right)^{2} \end{bmatrix}}} & \left( {{Eq}.\mspace{14mu} 32} \right) \\ {E_{\min} = {E\left( {r_{L},r_{a},r_{b}} \right)}} & \left( {{Eq}.\mspace{14mu} 33} \right) \end{matrix}$

Then for each iteration, let

$\begin{matrix} {g^{(k)} = \begin{pmatrix} {{\partial E}/r_{L}} \\ {{\partial E}/r_{a}} \\ {{\partial E}/r_{b}} \end{pmatrix}} & \left( {{Eq}.\mspace{14mu} 34} \right) \\ {{F(k)} = {\begin{pmatrix} {\frac{\partial^{2}E}{\partial^{2}r_{L}}\frac{\partial^{2}E}{{\partial r_{L}}{\partial r_{a}}}\frac{\partial^{2}E}{{\partial r_{L}}{\partial r_{b}}}} \\ {\frac{\partial^{2}E}{{\partial r_{a}}{\partial r_{L}}}\frac{\partial^{2}E}{\partial^{2}r_{a}}\frac{\partial^{2}E}{{\partial r_{a}}{\partial r_{b}}}} \\ {\frac{\partial^{2}E}{{\partial r_{b}}{\partial r_{L}}}\frac{\partial^{2}E}{{\partial r_{b}}{\partial r_{a}}}\frac{\partial^{2}E}{\partial^{2}r_{b}}} \end{pmatrix}.}} & \left( {{Eq}.\mspace{14mu} 35} \right) \end{matrix}$

Accordingly,

r _(k+1) =r ^((k)) −F(r ^((k)))⁽⁻¹⁾ g ^((k)).  (Eq. 36)

For every r^((k+1)), the weighting factor w_(j)'s are updated and the new function E(r) is obtained. The iteration stops when E(r) does not improve.

The algorithms for RBF data fitting may depend on the selection of norms. For color conversions, the norm may be selected considering the visual tolerance of the colors in the CIELab color space and the source of errors in the color conversion process. For example, different norms should be selected for differential color regions, such as

L₁ norm:

$\begin{matrix} {{x}_{1} = {\sum\limits_{i = 1}^{m}{x_{i}}}} & \left( {{Eq}.\mspace{14mu} 37} \right) \end{matrix}$

L₂ norm:

$\begin{matrix} {{x}_{2} = \sqrt{\sum\limits_{i = 1}^{m}x_{i}^{2}}} & \left( {{Eq}.\mspace{14mu} 38} \right) \end{matrix}$

L_(∞) norm:

$\begin{matrix} {{x}_{\infty} = {\max\limits_{1 \leq x \leq m}{{x_{i}}.}}} & \left( {{Eq}.\mspace{14mu} 39} \right) \end{matrix}$

The relationship of errors via L₁ norm, L₂ norm, and L_(∞) may be stated as:

For all e in C[a; b] the inequalities,

∥e∥ ₁≦(b−a)^(1/2) ∥e∥ ₂≦(b−a)∥e∥ _(∞)  (Eq. 40)

hold.

This theorem indicates that it small errors are found via L_(∞) norm, then the errors via L₁ norm and L₂ norm are also small. Thus, [a, b]=[0, 1], and the CIELab color space may be heuristically chosen in three regions:

1. Neutral region: 0≦C*_(ab)≦7; 2. Medium color region: 7<C*_(ab)≦30; and 3. Saturated color region: C*_(ab)>30, where C*_(ab) is the chroma of colors in CIELab color space.

For the neutral color region, the visual color difference tolerance is relatively small. Error from the measurements and machine can affect the accuracy of color conversion. The L₁ norm is chosen since it is least sensitive to the outliers. For the medium color range, the color difference tolerance is larger and the errors from the measurements and machine play less of a role affecting the quality of the color conversions. However, control of the color conversion error should be less than a magnitude of 2 so the color difference is less visible. The L₂ norm is chosen to minimize the average error. For the saturated color regions, the errors are generally small compared to the color conversion error of gamut mapping and it is unlikely for the HVS to detect any errors with magnitude less than 2. To minimize the maximum error of the color conversion, the L_(∞) norm is generally chosen.

As mentioned, the continuity and smoothness are important to the quality of the color conversion. When adjacent clusters or two adjacent regions have different mapping functions, data in the adjacent clusters may be included in the current cluster. Overlapping ranges specified as δ's for each region are optimized. The δ values are the distances calculated in the CIELab color space. The data regions partitioned for the L₁, L₂, and L_(∞) approximation are:

1. L₁ approximation: 0≦C*_(ab)<7+δ_(n). Intended range: 0≦C*_(ab)≦7;

2. L₂ approximation: 7−δ_(ml)<C*_(ab)<30+δ_(mu). Intended range: 7≦C*_(ab)≦30;

3. L_(∞) approximation: C*_(ab)>30−δ_(s). Intended range: C*_(ab)≧30,

where δ_(n) is the extended region for the L₁ norm, δ_(ml) and δ_(mu) are the upper and lower extended regions for the L₂ norm, and δ_(s) is the extended region for the L_(∞) norm. For the CMYK to CIELab conversion, the domain space is the undivided CMYK color space. For the Cielab to CMY conversion, the domain space is the CIELab color space divided into three regions based on the selected set of training data described above. Although shown and described with a particular range, the invention is not intended to be so limited. For example, the approximation may be varied as desired to change the granularity of the approximation.

Validation techniques, such as “cross validation” and “bootstraps”, may be used for model accuracy estimation and model selection. Among the cross validation techniques, the “K-fold cross” validation and the “leave-one-out” cross validation may be used. Using the cross validation technique, the absolute accuracy and the bias of variances should be balanced.

In general, several thousand data points are selected for building a color conversion model. A relatively large K number may be chosen to ensure the model accuracy. Different from other cross validation applications, the evaluation of color conversion models involves both numerical analysis and psychophysics analysis. For the numerical analysis, the test data set is collected spanning the input color space. For the perceptual analysis, a number of images may be selected in a visual experiment.

L₁, L₂, and LX approximations may be used to select the K value to the color conversion. For the L₂ approximation, the mean error may be used as the measure for the model selection. And, for the L₁ and L_(∞) approximations, the L₁ error and maximum error may be calculated instead, respectively. In any case, the variance as a function of the K value may be obtained to decide the K value. The cross validation technique is described using CMYK to CIELab conversion as an example. First, the modeling set is divided into K-fold partition data sets. For each of K experiments, K−1 folds are used for training with the remaining one for testing. Then, K-fold cross validation is applied to determine the best K value and model. The mean error decreases with the increase of the K value for K<10 and it stays roughly at the same level for K>15. On the other hand, the variance of the mean error stays relatively flat for K<15 and increases with the increase of the K value. Although the maximum error decreases and the mean error stays relatively flat for K≧15, the K value for the least squares approximation for the color conversion is generally 15 due to the increase of the variance.

The CMYK to CIELab conversion may be performed using the RBF data fitting algorithm in the L₂ norm. In this regard, data may be clustered using the combination of the LBG algorithm and the covariance matrix algorithm. Generally, there are two optimization stages. Initially, a relatively good starting function and radius for the R4→R3 optimization is sought. Then, models of R4→R for the CMYK to L*, a*, b* conversions are sought.

In the next modeling stage, ƒ_(L), ƒ_(a), and ƒ_(b) are set to multiquadratic with initial r_(L), r_(a), and r_(b). The multidimensional Newton's method is then applied to optimize these parameters. For the CIELab to CMY conversion, both data range (δ_(ml) and δ_(mu)) for the least squares approximation and the color conversion model are optimized by letting δ_(ml)=0, δ_(mu)=0, and k=0, such that s_(k)={(L,a,b)_(jk)|7≦C*_(abjk)≦30}, and initiating the clustering algorithm to find the clusters. Then, R3→R is optimized for the CIELab to C, M, and Y conversions, respectively. The multidimensional Newton's method of R3→R3 is performed for the CIELab to CMY conversion and the mean error and maximum error are recorded.

Afterwards, let δ_(ml)=2.5, δ_(mu)=2.5, and k=0, such that s_(k)={(L,a,b)_(jk)|7−2.5k≦C*_(abjk)≦30+2.5k,k=1,2, . . . ,} repeat the optimization process stopping when all data points are included in s_(k). The least square errors are obtained by calculating the differences between the original values and the estimated values. The optimized models indicate the Gaussian function for the cyan component, the inverse multiquadratic for the magenta component, and the multiquadratic function for the yellow component, thereby providing inside gamut color conversions from CIELab to CMY, g₀, and CMY to CIELab, f₀.

Overdetermined Linear Systems in the L_(∞) Sense

In saturated color areas, the perceptual color difference tolerance is generally higher than the magnitude of machine and instrument errors. Data outliers do not play a significant role in the color conversion. The maximum error is weighted in the color conversion quality assessments. Thus, it is desirable to reduce the maximum approximation error by solving the problem via the L_(∞) approximation, also referred as the Chebyshev norm. From norm inequalities if L_(∞) norm is controlled, then the L₁ and L₂ error is generally small.

Many algorithms have been developed for this type of application function approximation using the L_(∞) norm that are locally equivalent to the exchange algorithm. However, there is limited research on scattered data fitting techniques using the L_(∞) norm especially for the large sparse problems. Large sparse problems are often converted to linear programming problems using the linearized sub-problem as the basis. Among these algorithms, the Barrodale and Phillips dual algorithm (the “BP” algorithm) and the Bartels, Conn, and Li primal algorithm (the “BC” algorithm) are suitable algorithms for solving overdetermined linear systems Rn→R in the Chebyshev norm.

The BP and BC algorithms may be suitable in view of the similarity of the formulations with the RBF approach. As the BP and BC algorithms are generally used in R^(n)→R, the extension of the BP dual algorithm may be extended to R3→R3 for color conversion from CIELab to CMY. In this regard, the Chebyshev norm may be described by letting

Ax=b,  (Eq. 41)

where A=[a₁, a₂, . . . , a_(n)]εRmxm, m>n≧2, and b^(T)=[β₁, β₂, . . . , β_(m)]εRm. Then, the vector xεRn is determined such that

$\begin{matrix} {{{{Ax} - b}}_{\infty} = {\max\limits_{1 \leq i \leq m}{{{a_{i}^{T}x} - \beta_{i}}}}} & \left( {{Eq}.\mspace{14mu} 42} \right) \end{matrix}$

is minimized. L is then defined as a linear space spanned by the functions {φ_(i)}

L=<φ ₁(z),φ₂(z), . . . , φ_(n)(z)>,  (Eq. 43)

where each φ_(i) is continuous on [a,b],j=1, . . . , n. A function φεL may then be expressed as

$\begin{matrix} {{{\varphi (z)} = {{\sum\limits_{j = 1}^{n}{\alpha_{j}{\varphi_{j}(z)}}} \in L}},} & \left( {{Eq}.\mspace{14mu} 44} \right) \end{matrix}$

where zε[a, b]. The RBF is a relatively good example of such a space. Given m>n≧2, data points t(z_(i); y_(i)); i=1; 2; . . . ; m), α=(α₁, . . . , α_(n)) is determined to minimize

$\begin{matrix} {{e\left( {\alpha_{1},\ldots \mspace{11mu},\alpha_{n}} \right)} = {{y_{i} - {\sum\limits_{j = 1}^{n}{\alpha_{j}{\varphi_{j}\left( z_{i} \right)}}}}}_{\infty}} & \left( {{Eq}.\mspace{14mu} 45} \right) \\ {\mspace{146mu} {= {\max\limits_{1 \leq i \leq m}{{{y_{i} - {\sum\limits_{j - 1}^{n}{\alpha_{j}{\varphi_{j}\left( z_{i} \right)}}}}}.}}}} & \left( {{Eq}.\mspace{14mu} 46} \right) \end{matrix}$

Then,

$\begin{matrix} {x = {\alpha = \left\{ {a_{1},a_{2},\ldots \mspace{11mu},\alpha_{n}} \right\}}} & \left( {{Eq}.\mspace{14mu} 47} \right) \\ {A = \begin{pmatrix} {\varphi_{1}\left( z_{1} \right)} & {\varphi_{2}\left( z_{1} \right)} & \ldots & {\varphi_{n}\left( z_{1} \right)} \\ \ldots & \; & \; & \; \\ \ldots & \; & \; & \; \\ {\varphi_{1}\left( z_{m} \right)} & {\varphi_{2}\left( z_{m} \right)} & \ldots & {\varphi_{n}\left( z_{m} \right)} \end{pmatrix}} & \left( {{Eq}.\mspace{14mu} 48} \right) \end{matrix}$

and

b=(y ₁ y ₂ . . . y _(m))^(T).  (Eq. 49)

Next, the L_(∞) approximation may be converted to a standard form of the linear programming. First, however, the primal form of the problem is established by letting ξ denote the maximum error. For example,

$\begin{matrix} {{\xi = {\max\limits_{1 \leq i \leq m}{e_{i}\left( {\alpha_{1},\alpha_{2},\ldots \mspace{11mu},\alpha_{n}} \right)}}},} & \left( {{Eq}.\mspace{14mu} 50} \right) \end{matrix}$

where e is defined in Eq. 41.

The primal problem may then be formulated by minimizing ξ subject to

$\begin{matrix} {= {{\xi + {\sum\limits_{j - 1}^{n}{\alpha_{j}{\varphi_{j}\left( z_{i} \right)}}}} \geq {y_{i}\mspace{14mu} {and}}}} & \left( {{Eq}.\mspace{14mu} 51} \right) \\ {{= {{\xi - {\sum\limits_{j - 1}^{n}{\alpha_{j}{\varphi_{j}\left( z_{i} \right)}}}} \geq {- y_{i}}}},} & \left( {{Eq}.\mspace{14mu} 52} \right) \end{matrix}$

for every i where

ξ≧0  (Eq. 53)

and

α₁,α₂, . . . , α_(n) are unrestricted.  (Eq. 54)

The standard form of the corresponding dual linear program may be transformed from the above primal linear program via

$\begin{matrix} {\max {\sum\limits_{i = 1}^{m}{\left( {{\underset{\_}{\sigma}}_{i} - {\underset{\_}{\tau}}_{i}} \right)y_{i}}}} & \left( {{Eq}.\mspace{14mu} 55} \right) \end{matrix}$

subject to

$\begin{matrix} {{\sum\limits_{i = 1}^{m}{\left( {{\underset{\_}{\sigma}}_{i} - {\underset{\_}{\tau}}_{i}} \right)\varphi \; j_{i}}} = 0} & \left( {{Eq}.\mspace{14mu} 56} \right) \\ {{\sum\limits_{i = 1}^{m}\left( {{\underset{\_}{\sigma}}_{i} + {\underset{\_}{\tau}}_{i}} \right)} \leq 1} & \left( {{Eq}.\mspace{14mu} 57} \right) \end{matrix}$

and

σ _(i),τ _(i)≧0,i=1,2, . . . m,  (Eq. 58)

where φ_(ji)=φ_(j)(z_(i)).

To understand the BP dual algorithm, an understanding of the exchange algorithm and related theorems which almost all Chebyshev approximation algorithms are based is helpful. In this regard, A may be defined as a linear subspace of C [a,b], fεC [a, b], and L is any closed subset of [a, b], pεA, such that

e*(x)=f(x)−p*(x)  (Eq. 59)

and

L _(M) ={xεL|e*(x) obtains extreme values}.  (Eq. 60)

p* provides for a minimax solution to be approximation from A to ƒ, if ∀pεA, such that

e*(x)p(x)=[ƒ(x)−p*(x)]p(x)>0,∀xεL _(M)  (Eq. 61)

(e.g., e*(x) and p(x) are of same signs at all extreme points).

In this regard, the “Haar” condition defines a class of functions upon which the Exchange Algorithm can be applied. For example, A may be defined as an (n+1) dimensional linear subspace of C[a, b]. A satisfies the Haar condition if ∀pεA, the number of roots of the equation

{p(x)=0;a≦x≦b}  (Eq. 62)

is less than the dimension of A.

A derived condition states that if {ξ_(i)=1; 2, . . . , k}⊂[a, b], k≦n then ∃pεA, such that p changes signs at ξ_(i): i=1; 2, . . . , k and p has no other roots. The RBFs in the form of Eq. 11 meet the Harr condition because of the linear part in the equation. Then, let A be an (n+1)-dimensional linear subspace of C[a, b] that satisfies the Haar condition and let ƒ be any function in C[a, b]. Then p* is the minimax approximation from A to ƒ, if there exists (n+2) points {ξ*_(i)=0; 1; . . . ; n+1}, such that the conditions

a≦ξ* ₀<ξ*₁< . . . <ξ*_(n+1) ≦b,  1.

|ƒ(ξ*_(i))−p*(ξ*_(i))|=∥ƒ−p*∥_(∞) ,i=0,1, . . . , n+1 and  2.

ƒ(ξ*_(i+1))−p*(ξ*_(i+1))=−[ƒ(ξ*_(i))−p*(ξ*_(i))],i=0, . . . , n, alternative signs]] are obtained.  3.

Then, let A be a finite-dimension subspace of C[a, b] that satisfies the Haar condition and let {x_(i): i=1; 2; . . . ; m} be a set of distinct points from [a, b], where m is not less than the dimension of A. For any ƒ in C[a, b], let the one-point exchange algorithm be applied to calculate the element of A that minimizes expression

$\begin{matrix} {{\max\limits_{{i - 1},2,{\ldots \mspace{11mu} m}}{{{f\left( x_{i} \right)} - {p\left( x_{i} \right)}}}},{p \in A}} & \left( {{Eq}.\mspace{14mu} 63} \right) \end{matrix}$

Then, the required approximation to ƒ is obtained in a finite number of iterations.

In color conversion may compose a set of discrete data points. Thus, the Exchange Algorithm may be based on the Minimax Characterization Theorem (MCT) as shown in FIG. 4. For each iteration k, the MCT adjusts a reference {ξ*_(i): i=0; 1; . . . ; n+1} and a trial approximation p_(k) is determined. On a discrete set, ƒ may be known on a set of points {x_(i); i=1; 2; : : : ; m}. To determine a trial approximation p_(k) in each iteration, let

h=ƒ(ξ₀)−p _(k)(τ₀)  (Eq. 64)

and solve

ƒ(ξ_(i))−p(ξ_(i))=(−1)^(i) h,i=+0,1,2, . . . , n+1  (Eq. 65)

The iteration stops when p_(k) satisfies the conditions in the MCT.

For solving the dual problem, the BP dual algorithm is reviewed. The BP dual algorithm is based on the standard form of the simplex method composed of three stages and introduces artificial variables α_(j) in Eq. 55 and a slack variable ξ in Eq. 56. The above dual problem thereby becomes

$\begin{matrix} {\max {\sum\limits_{i = 1}^{m}{\left( {{\underset{\_}{\sigma}}_{i} - {\underset{\_}{\tau}}_{i}} \right)y_{i}}}} & \left( {{Eq}.\mspace{14mu} 66} \right) \end{matrix}$

subject to

$\begin{matrix} {{{{\sum\limits_{i = 1}^{m}{\left( {{\underset{\_}{\sigma}}_{i} - {\underset{\_}{\tau}}_{i}} \right)\varphi_{j_{i}}}} + {\underset{\_}{\alpha}}_{i}}\; = 0},{j = 1},2,\ldots \mspace{11mu},n} & \left( {{Eq}.\mspace{14mu} 67} \right) \\ {{{\sum\limits_{i = 1}^{m}\left( {{\underset{\_}{\sigma}}_{i} + {\underset{\_}{\tau}}_{i}} \right)} + \underset{\_}{\xi}} = 1} & \left( {{Eq}.\mspace{14mu} 68} \right) \\ {\xi,{\underset{\_}{\alpha}}_{i},{{- {\underset{\_}{\tau}}_{i}} \geq 0},{j = 1},2,\ldots \mspace{11mu},{m.}} & \left( {{Eq}.\mspace{14mu} 69} \right) \end{matrix}$

Denoting the column vectors formed by the coefficients of σ _(i), α _(i), τ _(i), and ξ by s _(i), t _(i), a _(j), and w, then

si+t _(i)=2w  (Eq. 70)

Then, let b denote the right-hand side of Eq. 52 such that

b=w=(0 0 . . . 0 1)^(T) =e _(n+1) ^(T).  (Eq. 71)

The first two stages of the BP dual algorithm include determining an efficient starting point for the third stage, which is equivalent to the exchange algorithm for the linear minimax approximation. In doing so, the BP dual algorithm uses the table below:

Then, Φ is denoted as an m×n matrix:

$\begin{matrix} {\Phi = {\begin{pmatrix} \varphi_{1,1} & \varphi_{2,1} & \ldots & \varphi_{n,1} \\ \varphi_{1,2} & \varphi_{2,2} & \ldots & \varphi_{n,2} \\ \ldots & \; & \; & \; \\ \ldots & \; & \; & \; \\ \varphi_{1,m} & \varphi_{2,m} & \ldots & \varphi_{n,m} \end{pmatrix}\mspace{14mu} {and}}} & \left( {{Eq}.\mspace{14mu} 72} \right) \end{matrix}$

the rank of Φ is denoted as k.

In the first stage, the first k simplex iterations are performed. The algorithm finds the maximum residual at each iteration and reduce it to zero. The nonbasic variables that can enter the basis to become basic variables are σ _(i)'s, and basic variables that can leave the basis to become nonbasic variables are restricted to α _(i)s. Basic variable ξ remains zero in the basis. At the end of stage 1, there are k residues at zero. σ _(t) that enters the basis corresponds to that with the largest absolute reduced cost (e.g., choosing the largest residual in the primal problem). The pivot may be selected from this column as the largest absolute value corresponding to an α _(j) in the basis. At the t^(th) iteration, with the corresponding σ _(t) entering the basis, there are t zero residuals. For example,

$\begin{matrix} {{y_{t} - {\sum\limits_{j = 1}^{n}{\alpha_{j}{\varphi_{j}\left( z_{t} \right)}}}} = {\xi = 0.}} & \left( {{Eq}.\mspace{14mu} 73} \right) \end{matrix}$

In the second stage, the k+1)^(th) simplex iteration is performed and the variable ξ is forced to leave the basis. This means the error ξ in the primal problem is increased above zero. At the end of the second stage, the k+1 residuals of magnitude equal to resulting value of ξ. The (k+1)^(th) pivotal column may be chosen also corresponding to the largest absolute marginal cost (e.g., equivalent to choosing the largest residual in the primal problem). The σ_(j)'s and τ_(j)'s are exchanged if the marginal cost is negative. If there are positive values in the first n elements in the pivotal column, these values are changed to negative by adding twice of the corresponding rows to the (n+1)^(th) pivotal row and changing the sign of those original rows.

The third stage is equivalent to the exchange algorithm for a linear minimax approximation. The pivotal column may be chosen corresponding to the most negative marginal cost or a ratio selection rule. The iteration continues until all marginal cost is nonnegative. Every iteration may increase the value of ξ until ξ=e*, where e* is the minimum value in Eq. 45.

Alternatively, the BC algorithm may be used to determine the starting point for color conversion. In doing so, the primal formulation in Eq. 45 is rewritten by letting

$\begin{matrix} {c_{0} = \left( {1\mspace{14mu} 0\mspace{14mu} 0\mspace{14mu} 0\mspace{14mu} \ldots \mspace{14mu} 0} \right)_{n + 1}^{T}} & \left( {{Eq}.\mspace{14mu} 74} \right) \\ {c_{i} = \begin{pmatrix} 1 \\ {- a_{i}} \end{pmatrix}_{n + 1}} & \left( {{Eq}.\mspace{14mu} 75} \right) \\ {c_{m + i} = \begin{pmatrix} 1 \\ {+ a_{i}} \end{pmatrix}_{n + 1}} & \left( {{Eq}.\mspace{14mu} 76} \right) \\ {\delta_{i} = {- y_{i}}} & \left( {{Eq}.\mspace{14mu} 77} \right) \\ {{\delta_{m + i} = y_{i}},{i = 1},2,\ldots \mspace{11mu},m,{and}} & \left( {{Eq}.\mspace{14mu} 78} \right) \\ {v = \begin{pmatrix} \xi \\ \alpha \end{pmatrix}_{n + 1}} & \left( {{Eq}.\mspace{14mu} 79} \right) \end{matrix}$

The above primal problem is then converted to an unconstrained optimization problem via a piecewise linear penalty function. For example, let μ>0 be a fixed parameter and define

$\begin{matrix} {{p\left( {{vi},\mu} \right)} = {{\mu \; c_{0}^{T}v} - {\sum\limits_{j = 1}^{2\; m}{\min \left( {0,{{c_{j}^{T}v} - \delta_{j}}} \right)}}}} & \left( {{Eq}.\mspace{14mu} 80} \right) \end{matrix}$

For any arbitrary vεRn+1, Eq. 80 can be expanded into

$\begin{matrix} {{p\left( {v,\mu} \right)} = {{\mu \; c_{0}^{T}v} - {\sum\limits_{j \in I^{0}}^{\;}{\min \left( {0,{{c_{j}^{T}v} - \delta_{j}}} \right)}}}} & \left( {{Eq}.\mspace{14mu} 81} \right) \\ {- {\sum\limits_{j \in I^{+}}^{\;}{\min \left( {0,{{c_{j}^{T}v} - \delta_{j}}} \right)}}} & \left( {{Eq}.\mspace{14mu} 82} \right) \\ {{- {\sum\limits_{j \in I^{-}}^{\;}{\min \left( {0,{{c_{j}^{T}v} - \delta_{j}}} \right)}}},} & \left( {{Eq}.\mspace{14mu} 83} \right) \end{matrix}$

where

I ⁰ =I ⁰(v)={j|c _(j) ^(T) v=δ _(j) }={j ₁ ,j ₂ , . . . , j _(k)},

I ⁺ =I ⁺(v)={j|c _(j) ^(T) v>δ _(j)},and

I ⁻ =I ⁻(v)={j|c _(j) ^(T) v<δ _(j)}.

For any dε

^(n+1) and any λ≧0 sufficiently small,

$\begin{matrix} {{p\left( {{v + {\lambda \; d}},\mu} \right)} = {{p\left( {v + \mu} \right)} + {\lambda \begin{bmatrix} {{\mu \; c_{0}^{T}d} - {\sum\limits_{j \in I^{-}}^{\;}{c_{j}^{T}d}} -} \\ {\sum\limits_{j \in I^{0}}^{\;}{\min \left( {0,{c_{j}^{T}d}} \right)}} \end{bmatrix}}}} & \left( {{Eq}.\mspace{14mu} 84} \right) \\ {\mspace{130mu} {{{= {{p\left( {v + \mu} \right)} + {\lambda \; h^{T}d} + {\lambda {\sum\limits_{j \in I^{0}}^{\;}{\sigma_{j}^{-}c_{j}^{T}d}}}}},{where}}{{h = {{\mu \; c_{0}} - {\sum\limits_{j \in I^{-}}^{\;}c_{j}}}},{and}}{\sigma_{j}^{-} = {{\sigma_{j}^{-}(d)} = \left\{ {{\begin{matrix} 0 & {{{if}\mspace{14mu} c_{j}^{T}d} \geq 0} \\ 1 & {{{{if}\mspace{14mu} c_{j}^{T}d} < 0},} \end{matrix}j} \in {I^{0}.}} \right.}}}} & \left( {{Eq}.\mspace{14mu} 85} \right) \end{matrix}$

Then, define the matrix

N=[c_(j1),c_(j2), . . . , c_(jk)]  (Eq. 86)

and let

P=I−N(N ^(T) N)⁻¹ N ^(T).  (Eq. 87)

The matrix P is the orthogonal projector onto the null space of N^(T). To define search direction d such that p(v+λd, μ)<p(v, μ),

d=−Ph  (Eq. 88)

is established. When Ph≠0, apply d defined in Eq. 88 to Eq. 85 such that

$\begin{matrix} \begin{matrix} {{c_{j}^{T}d} = {{- c_{j}^{T}}{Ph}}} \\ {= {{- c_{j}^{T}}{PPh}}} \\ {= {{- \left( {Pc}_{j} \right)^{T}}{Ph}}} \\ {= 0} \end{matrix} & \left( {{Eq}.\mspace{14mu} 89} \right) \end{matrix}$

Since P is the orthogonal projector onto the null space of N^(T). The Eq. 84 becomes:

p(v+λd,μ)= p(v,μ)+λh ^(T) d  (Eq. 90)

h^(T)d<0, and  (Eq. 91)

p(v+λd,μ)< p(v,μ).  (Eq. 92)

Thus, d serves as a descent direction for the function p.

When Ph=0, assuming columns of N are linearly independent, then

$\begin{matrix} {h = {\sum\limits_{i = 1}^{k}{\eta_{i}{c_{ji}.}}}} & \left( {{Eq}.\mspace{14mu} 93} \right) \end{matrix}$

and Eq. 84 becomes

$\begin{matrix} {{p\left( {{v + {\lambda \; d}},\mu} \right)} = {{p\left( {v,\mu} \right)} + {\lambda {\sum\limits_{i = 1}^{k}{\left\lbrack {\eta_{i} + \sigma_{ji}^{-}} \right\rbrack c_{j}^{T}{d.}}}}}} & \left( {{Eq}.\mspace{14mu} 94} \right) \end{matrix}$

In the sub case of a: η_(i)*<0, for some i*εI⁰, d is chosen such that

c_(j) ^(T)d=0,jεI⁰,j≠i.  (Eq. 95)

c _(i) ^(T) *d=1  (Eq. 96)

This implies that σ_(j) ⁻=0<0, jεI⁰. Because

h ^(T) d=η _(i) *c _(i) ^(T) *d=η<0,  (Eq. 97)

then

p(v+λd,μ)<p(v,μ).  (Eq. 98)

In the sub case of b: η_(i)*<0 for all i*εI⁰, if no constraints are violated, it is the optimal. Otherwise, reduce μ and start another iteration. For example,

{choose step size A}

$\begin{matrix} {{\Lambda^{(1)}\overset{def}{=}\left\{ {{\left. \lambda \middle| \lambda \right. = \frac{\delta_{j} - {c_{j}^{T}v}}{c_{j}^{T}d}},{\lambda > 0},{j \in {I^{+}\bigcup I^{-}}}} \right\}},} & \left( {{Eq}.\mspace{14mu} 99} \right) \\ {\lambda = {\min \; \Lambda^{(1)}}} & \left( {{Eq}.\mspace{14mu} 100} \right) \end{matrix}$

and update v=v+λd.

As an example, the L_(∞) norm approximation may be applied to color conversions from CMYK to CIELab and CIELab to CMY. The formulation of this color conversion in the primal form is described in the following using the CIELab to CMY color conversion by minimizing

$\begin{matrix} {\xi = {\xi_{c} + \xi_{m} + {\xi_{y}\mspace{14mu} {subject}\mspace{20mu} {to}}}} & \left( {{Eq}.\mspace{14mu} 101} \right) \\ {{\xi_{c} + {\sum\limits_{j = 1}^{n + 4}\; {\alpha_{cj}{F_{cj}\left( z_{i} \right)}}}} \geq c_{i}} & \left( {{Eq}.\mspace{14mu} 102} \right) \\ {{\xi_{c} - {\sum\limits_{j = 1}^{n + 4}\; {\alpha_{cj}{F_{cj}\left( z_{i} \right)}}}} \geq {- c_{i}}} & \left( {{Eq}.\mspace{14mu} 103} \right) \\ {{\xi_{m} + {\sum\limits_{j = 1}^{n + 4}\; {\alpha_{mj}{F_{mj}\left( z_{i} \right)}}}} \geq m_{i}} & \left( {{Eq}.\mspace{14mu} 104} \right) \\ {{\xi_{m} - {\sum\limits_{j = 1}^{n + 4}\; {\alpha_{mj}{F_{mj}\left( z_{i} \right)}}}} \geq {- m_{i}}} & \left( {{Eq}.\mspace{14mu} 105} \right) \\ {{\xi_{y} + {\sum\limits_{j = 1}^{n + 4}\; {\alpha_{yj}{F_{yj}\left( z_{i} \right)}}}} \geq y_{i}} & \left( {{Eq}.\mspace{14mu} 106} \right) \\ {{{\xi_{y} - {\sum\limits_{j = 1}^{n + 4}\; {\alpha_{yj}{F_{yj}\left( z_{i} \right)}}}} \geq {- y_{i}}},} & \left( {{Eq}.\mspace{14mu} 107} \right) \end{matrix}$

where

ξ≧0,

α₁,α₂, . . . α_(n+4) are unrestricted,

z_(i)={L,a,b}i,i=1, . . . , m,

F _(ji)=[1,z _(i),φ₁(z _(i)),φ₂(z _(i)), . . . , φ_(n)(z _(i))]_(4+n),

φ_(ji)=φ_(j)(z _(i)),

φ_(j) is one of the RBFs, n is the number of clusters, and m is the number of color data points.

The problem may then be rewritten by minimizing c₀ ^(T)v subject to

$\begin{matrix} {\mspace{79mu} {{{{{c_{j}^{T}v} \geq \delta_{j}},{j = 1},2,{\ldots \mspace{14mu} 6m\mspace{14mu} {where}}}\mspace{79mu} {c_{0} = \left( {1\mspace{14mu} 0\mspace{14mu} \ldots \mspace{14mu} 1\mspace{14mu} 0\mspace{14mu} \ldots \mspace{14mu} 1\mspace{14mu} 0\mspace{14mu} \ldots \mspace{14mu} 0} \right)_{{3n} + 15}^{T}}{v = \left( {\xi_{c}\mspace{14mu} \alpha_{c1}\mspace{14mu} \ldots \mspace{14mu} \alpha_{cn}\mspace{14mu} \xi_{m}\mspace{14mu} \alpha_{m1}\mspace{14mu} \ldots \mspace{14mu} \alpha_{mn}\mspace{14mu} \xi_{y}\mspace{14mu} \alpha_{y1}\mspace{14mu} \ldots \mspace{14mu} \alpha_{yn}} \right)_{{3n} + 15}^{T}}\mspace{79mu} {c_{i} = \left( {1\mspace{14mu} - {\alpha_{ci}\mspace{14mu} \ldots \mspace{14mu} 0\mspace{14mu} \ldots \mspace{14mu} 0_{{2n} + 8}}} \right)_{{3n} + 15}^{T}}\; \mspace{79mu} {c_{m + i} = \left( {1\mspace{14mu} \alpha_{ci}\mspace{14mu} \ldots \mspace{14mu} 0\mspace{14mu} \ldots \mspace{14mu} 0_{{2n} + 8}} \right)_{{3n} + 15}^{T}}\mspace{79mu} {c_{{2m} + i} = \left( {{0\mspace{14mu} \ldots \mspace{14mu} 0_{n + 4}\mspace{14mu} \ldots \mspace{14mu} 1\mspace{14mu} \ldots}\mspace{14mu} - {\alpha_{mi}\mspace{14mu} 0\mspace{14mu} \ldots \mspace{14mu} 0_{n + 4}}} \right)_{{3n} + 15}^{T}}\mspace{79mu} {c_{{3m} + i} = \left( {0\mspace{14mu} \ldots \mspace{14mu} 0_{n + 4}\mspace{14mu} \ldots \mspace{14mu} 1\mspace{14mu} \alpha_{mi}\mspace{14mu} 0\mspace{14mu} \ldots \mspace{14mu} 0_{n + 4}} \right)_{{3n} + 15}^{T}}}\mspace{79mu} {c_{{4m} + i} = \left( {{0\mspace{14mu} \ldots \mspace{14mu} 0_{{2n} + 8}\mspace{14mu} 1}\mspace{14mu} - \alpha_{yi}} \right)_{{3n} + 15}^{T}}\mspace{79mu} {c_{{5m} + i} = \left( {0\mspace{14mu} \ldots \mspace{14mu} 0_{{2n} + 8}\mspace{14mu} 1\mspace{14mu} \alpha_{yi}} \right)_{{3n} + 15}^{T}}\mspace{79mu} {\delta_{i} = {- C_{i}}}\mspace{79mu} {\delta_{m + i} = C_{i}}\mspace{79mu} {\delta_{{2m} + i} = {- M_{i}}}\mspace{79mu} {\delta_{{3m} + i} = M_{i}}\mspace{79mu} {\delta_{{4m} + i} = {- Y_{i}}}\mspace{79mu} {{\delta_{{5m} + i} = Y_{i}},{and}}\mspace{79mu} {{i = 1},\ldots \mspace{14mu},{m.}}}} & \left( {{Eq}.\mspace{14mu} 108} \right) \end{matrix}$

The primal problem may then be divided into steps using the modified BC algorithm. For example, the primal problem with constraints may be converted to an unconstrained problem via a piecewise linear penalty function by letting μ>0 be a fixed parameter. Next,

$\begin{matrix} {{p\left( {v,\mu} \right)} = {{\mu \; c_{0}^{T}v} - {\sum\limits_{j = 1}^{6m}\; {\min \left( {0,{{c_{j}^{T}v} - \delta_{j}}} \right)}}}} & \left( {{Eq}.\mspace{14mu} 109} \right) \end{matrix}$

is defined. For any arbitrary vε

,

$\begin{matrix} {{{p\left( {v,\mu} \right)} = {{\mu \; c_{0}^{T}v} - {\sum\limits_{j \in I^{0}}\; {\min \left( {0,{{c_{j}^{T}v} - \delta_{j}}} \right)}} - {\sum\limits_{j \in I^{+}}\; {\min \left( {0,{{c_{j}^{T}v} - \delta_{j}}} \right)}} - {\sum\limits_{j \in I^{-}}\; {\min \left( {0,{{c_{j}^{T}v} - \delta_{j}}} \right)}}}},} & \left( {{Eq}.\mspace{14mu} 110} \right) \end{matrix}$

where

I ⁰ =I ⁰(v)={j|c _(j) ^(T) v=δ _(j) }={j ₁ ,j ₂ , . . . , j _(k)},

I ⁺ =I ⁺(v)={j|c _(j) ^(T) v>δ _(j)}, and

I ⁻ =I ⁻(v)={j|c _(j) ^(T) v<δ _(j)}.

Then, the sets I⁰, I⁺, and I⁻ are determined and the search direction and step size are updated.

As an example of color conversion in dual for CIELab to CMY,

$\begin{matrix} {{{Maximize}{\sum\limits_{i = 1}^{m}\; {\left( {{\underset{\_}{\sigma}}_{ci} - {\underset{\_}{\tau}}_{ci}} \right)c_{i}}}} + {\left( {{\underset{\_}{\sigma}}_{mi} - {\underset{\_}{\tau}}_{mi}} \right)m_{i}} + {\left( {{\underset{\_}{\sigma}}_{yi} - {\underset{\_}{\tau}}_{yi}} \right)y_{i}}} & \left( {{Eq}.\mspace{14mu} 111} \right) \end{matrix}$

subject to

$\begin{matrix} {{\sum\limits_{i = 1}^{m}\; {\left( {{\underset{\_}{\sigma}}_{ci} + {\underset{\_}{\tau}}_{ci}} \right)F_{C_{ji}}^{T}}} = 0} & \left( {{Eq}.\mspace{14mu} 112} \right) \\ {{\sum\limits_{i = 1}^{m}\; \left( {{\underset{\_}{\sigma}}_{ci} - {\underset{\_}{\tau}}_{ci}} \right)} \leq 1} & \left( {{Eq}.\mspace{14mu} 113} \right) \\ {{\sum\limits_{i = 1}^{m}\; {\left( {{\underset{\_}{\sigma}}_{Mi} + {\underset{\_}{\tau}}_{Mi}} \right)F_{M_{ji}}^{T}}} = 0} & \left( {{Eq}.\mspace{14mu} 114} \right) \\ {{\sum\limits_{i = 1}^{m}\; \left( {{\underset{\_}{\sigma}}_{Mi} - {\underset{\_}{\tau}}_{Mi}} \right)} \leq 1} & \left( {{Eq}.\mspace{14mu} 115} \right) \\ {{\sum\limits_{i = 1}^{m}\; {\left( {{\underset{\_}{\sigma}}_{Yi} - {\underset{\_}{\tau}}_{Yi}} \right)F_{Y_{ji}}^{T}}} = 0} & \left( {{Eq}.\mspace{14mu} 116} \right) \\ {{\sum\limits_{i = 1}^{m}\; \left( {{\underset{\_}{\sigma}}_{Yi} + {\underset{\_}{\tau}}_{Yi}} \right)} \leq 1} & \left( {{Eq}.\mspace{14mu} 117} \right) \end{matrix}$

and

σ _(i),τ _(i)≧0,i=1,2, . . . , m,  (Eq. 118)

where

z_(i)={L,a,b}_(i),i=1,2, . . . , m

F _(ji)=[1,z _(i),φ₁(z _(i)),φ₂(z _(i)), . . . , φ_(n)(z _(i))]_(n+4)

φ_(ji)=φ_(j)(z _(i)),

φ_(j) is one of the RBFs, n is the number of clusters, and m is the number of color data points.

The CIELab to CMY conversion may then be represented by the initial simplex table:

and the rank of the matrix F_(n×m), k is thereby determined.

Thereafter, the first step BP simplex algorithm may be performed via k simplex iterations for each of F_(Cn×m) for the cyan component, F_(Mn×m) for the magenta component, and F_(Yn×m) for the yellow component. The vectors s_(C) are allowed to enter the basis and the vectors α_(C) can leave the basis for F_(Cn×m). The vectors s_(M) are allowed to enter the basis and the vectors α_(M) can leave the basis for F_(Mn×m). The vectors s_(Y) are allowed to enter the basis and the vectors α_(Y) can leave the basis for Y_(n×m).

The second step of the BP simplex algorithm may be performed by forcing τ_(C), ξ_(M), and ξ_(Y) to leave the basis. The corresponding vectors entering the basis are sC, s_(M), and s_(Y) respectively. If the corresponding marginal cost of s_(C), s_(M), or s_(Y) is positive, then si is changed to t_(i) using

t _(i)=2w3_(n+4) −s _(i),

where

w _(C)=(0 0 . . . 0 1 0 0 0)^(T) =e _(3n+1) ^(T),

w _(M)=(0 0 . . . 0 0 1 0 0)^(T) =e _(3n+2) ^(T), and

w _(Y)=(0 0 . . . 0 0 0 1 0)^(T) =e _(3n+3) ^(T);

If s_(i) is the pivotal column, r is the row vector of pivotal row. For any j^(th) row vector r_(j) containing s_(ij)>0, where j≠3n+1 for F_(Cn×m), j≠3n+2 for F_(Mn×m), and j≠3n+3 for F_(Yn×m), change r to r+2r_(j) and change r_(j) to −r_(j). The third step of the BP simplex iterations may then be performed until all the marginal costs are non-negative.

Color Conversion Using the L₁ Norm Approximation

Color conversion in neutral color areas plays an important role in color reproduction quality. The human visual system (HVS) is sensitive to color changes in neutral color areas. Additionally, the color conversion errors due to machine stability, paper uniformity, and instrument accuracy are more profound relative to the human color difference tolerances. Thus, there are more chances for the existences of outliers using color conversion. A color conversion should minimize the negative impact of outliers. The L₁ norm approximation may assist in the region due to the robustness to outliers.

Among all the L₁ algorithms, the Barrodale and Roberts' primal algorithm (BR algorithm) is relatively robust. Similar to the problem setup for the L_(∞), the overdetermined linear system of the general L₁ approximation problem may be stated as follows:

Ax=b,  (Eq. 119)

where

A=[a ₁ ,a ₂ , . . . , a _(n)]ε

,(m>n≧2)

and

b^(T)=[β₁,β₂, . . . , β_(m)]ε

.

The L₁ norm approximation is used to determine vector xε

^(n) such that

$\begin{matrix} {{{{Ax} - b}}_{1} = {\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n}{{{a_{j}^{T}x_{i}} - \beta_{i}}}}}} & \left( {{Eq}.\mspace{14mu} 120} \right) \end{matrix}$

is minimized. In this regard, L is a linear space spanned by function φ_(i)

L=<φ ₁(z),φ₂(z), . . . , φ_(n)(z)>,  (Eq. 121)

where each φ_(i) is continuous on [a, b], i=1, . . . , n. The function φ may be expressed as

$\begin{matrix} {{{\varphi (z)} = {{\sum\limits_{i = 1}^{n}{\alpha_{i}{\varphi_{i}(z)}}} \in L}},} & \left( {{Eq}.\mspace{14mu} 122} \right) \end{matrix}$

where zε[a,b]⊂

.

Given m>n≧2, data points (z_(i), y_(i)), α=(α₁, . . . α_(n)) is determined to minimize

$\begin{matrix} {{e\left( {\alpha_{1}\mspace{14mu} \ldots \mspace{14mu} \alpha_{n}} \right)} = {{y_{i} - {\sum\limits_{j = 1}^{n}{\alpha_{j}{\varphi_{j}\left( z_{i} \right)}}}}}_{1}} & \left( {{Eq}.\mspace{14mu} 123} \right) \\ {= {\sum\limits_{j = 1}^{m}{{y_{j} - {\sum\limits_{j = 1}^{n}{\alpha_{j}{\varphi_{j}\left( z_{i} \right)}}}}}}} & \left( {{Eq}.\mspace{14mu} 124} \right) \end{matrix}$

The equation then becomes:

$\begin{matrix} {{x = a}} & \; \\ {{A = \begin{pmatrix} {\varphi_{1}\left( z_{1} \right)} & {\varphi_{2}\left( z_{1} \right)} & \ldots & {\varphi_{n}\left( z_{1} \right)} \\ \ldots & \; & \; & \; \\ \ldots & \; & \; & \; \\ {\varphi_{1}\left( z_{m} \right)} & {\varphi_{2}\left( z_{m} \right)} & \ldots & {\varphi_{n}\left( z_{m} \right)} \end{pmatrix}}b = \left( {y_{1}y_{2}\mspace{14mu} \ldots \mspace{14mu} \ldots \mspace{14mu} \ldots \mspace{14mu} y_{m}} \right)^{T}} & \left( {{Eq}.\mspace{14mu} 125} \right) \end{matrix}$

In the RBF form, the function becomes

$\begin{matrix} {{{\varphi (z)} = {\alpha_{0} + {\sum\limits_{p = 1}^{k}{\alpha_{p}{z(p)}}} - {\sum\limits_{j = {n + k}}^{n + k}{\alpha_{j}{\varphi \left( {{z - c_{j}}} \right)}}}}},} & \left( {{Eq}.\mspace{14mu} 126} \right) \end{matrix}$

where x=(α₀, α₁, . . . , α_(k), α_(k+1), . . . , α_(n+k)) and

$A = {\begin{pmatrix} 1 & {z(1)} & {z_{1}(2)} & {\ldots \;} & {z_{1}(k)} & {\varphi_{1}\left( z_{1} \right)} & {\varphi_{2}\left( z_{1} \right)} & \ldots & {\varphi_{n}\left( z_{1} \right)} \\ \ldots & \; & \; & \; & \; & \; & \; & \; & \; \\ \ldots & \; & \; & \; & \; & \; & \; & \; & \; \\ \ldots & \; & \; & \; & \; & \; & \; & \; & \; \\ 1 & {z_{m}(1)} & {z_{m}(2)} & \ldots & {z_{m}(k)} & {\varphi_{1}\left( z_{m} \right)} & {\varphi_{2}\left( z_{m} \right)} & \ldots & {\varphi_{n}\left( z_{m} \right)} \end{pmatrix}.}$

The L₁ and L_(∞) norms in C[a, b] and in R_(n) are not strictly convex. Accordingly, A is a linear subspace of either the L₁ or the L_(∞) normed linear space, as the uniqueness of best approximations from A to ƒ depends on properties of A and ƒ. p* in A may be the better L₁ approximation function to ƒ. A sign function is thus defined as:

$\begin{matrix} {{s^{*}(x)} = \left\{ \begin{matrix} {{- 1},} & {{f(x)} < {p^{*}(x)}} \\ {0,} & {{f(x)} = {p^{*}(x)}} \\ {1,} & {{f(x)} > {p^{*}(x)}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 127} \right) \end{matrix}$

A sufficient condition for the function p* is the L₁ approximation from A to ƒ which may be defined by letting A be a linear subspace of C[a, b], Abe any function in C[a, b], and p* be any element of A, such that the set

L={x:ƒ( x)=p*(x),a≦x≦b}  (Eq. 128)

is either empty or is composed of a finite number of intervals and discrete points. Then p* provides an L₁ norm approximation form A to ƒ, if the inequality

$\begin{matrix} {{\left. {\int_{a}^{b}{{s^{*}(x)}{p(x)}\ {x}}} \right\} } \leq {\int_{L}^{\;}{{{p(x)}}\ {x}\mspace{14mu} (4.7)}}} & \left( {{Eq}.\mspace{14mu} 129} \right) \end{matrix}$

is satisfied for all p in A, where s* is the function of Eq. 127.

For discrete L₁ approximation, let the function values {ƒ(x_(t)); t=1, 2, . . . m} and the fixed positive weights be {w_(t); t=1, 2, . . . , m}. A is a linear space of functions that may be defined on the point set {x_(t); t=1, 2, . . . , m}, and p* is any element of A. Then, C contains the points of {x_(t); t=1, 2, . . . , m} that satisfy the condition

p*(x _(t))=f(x _(t))  (Eq. 130)

and s* is the sign function

$\begin{matrix} {{s^{*}(x)} = \left\{ \begin{matrix} {{- 1},} & {{f\left( x_{t} \right)} < {p^{*}\left( x_{t} \right)}} \\ {0,} & {{f\left( x_{t} \right)} = {p^{*}\left( x_{t} \right)}} \\ {1,} & {{f\left( x_{t} \right)} > {p^{*}\left( x_{t} \right)}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 131} \right) \end{matrix}$

t=1, 2, . . . , m. p* becomes the function in A that minimizes the expression

$\begin{matrix} {{\sum\limits_{t = 1}^{m}{w_{t}{{{f\left( x_{t} \right)} - {p\left( x_{t} \right)}}}}},{p \in A}} & \left( {{Eq}.\mspace{14mu} 132} \right) \end{matrix}$

if the inequality

$\begin{matrix} {{{\sum\limits_{t = 1}^{m}{w_{t}{s^{*}\left( x_{t} \right)}{p\left( x_{t} \right)}}}} \leq {\sum\limits_{x_{t} \in L}^{m}{w_{t}{{p\left( x_{t} \right)}}}}} & \left( {{Eq}.\mspace{14mu} 133} \right) \end{matrix}$

holds for all p in A.

In order to improve the discrete L₁ approximation p*, the inequality in Eq. 133 is tested to be satisfied for every element p in A. An optimal function p* can be obtained by searching for suitable interpolation points in L. For example, let the function values {ƒ(x_(t)); t=1, 2, . . . , m} and the fixed positive weights be {w_(t); t=1, 2, . . . , m}. Then, A is a linear subspace of R^(m), where the component of each vector p in A have the values {p(x_(t)); t=1, 2, . . . , m}. An element p* in A minimizes Eq. 132, and the property that the zero vector is the element p in A that satisfies the conditions {p(x_(t))=0; x_(t)*L}, where the set L is defined in Eq. 128

With this is mind, the discrete L₁ approximation calculation becomes a linear programming problem. For example with φ={φ_(i); i=0, 1, . . . , n} as a basis of the space A of approximation, the Eq. 133 becomes

$\begin{matrix} {\sum\limits_{t = 1}^{m}{w_{t}{{y_{t} - {\sum\limits_{i = 0}^{n}{\lambda_{i}{\varphi_{i}\left( x_{t} \right)}}}}}}} & \left( {{Eq}.\mspace{14mu} 134} \right) \end{matrix}$

where y_(t) is the output value corresponding to each x_(t). Define the bounds {u_(t)≧0} and {v_(t)≧0} may be defined such that

$\begin{matrix} {{- v_{t}} \leq {y_{t} - {\sum\limits_{i = 0}^{n}{\lambda_{i}{\varphi_{i}\left( x_{t} \right)}}}} \leq u_{t}} & \left( {{Eq}.\mspace{14mu} 135} \right) \end{matrix}$

for {t=1, 2, . . . , m}. Then the linear programming problem is

$\begin{matrix} {{Minimize}{\sum\limits_{t = 1}^{m}{w_{t}\left( {u_{t} + v_{t}} \right)}}} & {\left( {{Eq}.\mspace{14mu} 136} \right)\mspace{14mu}} \end{matrix}$

subject to

$\begin{matrix} {{{- v_{t}} \leq {y_{t} - {\sum\limits_{i = 0}^{n}{\lambda_{i}{\varphi_{i}\left( x_{t} \right)}}}} \leq u_{t}},{t = 1},2,{\ldots \mspace{14mu} m}} & \left( {{Eq}.\mspace{14mu} 137} \right) \end{matrix}$ ut≧0

vt≧0

for t=1,2, . . . , m.

The algorithm may be described as:

$\begin{matrix} {{{f_{i} - {\sum\limits_{j = 1}^{n}{\alpha_{j}\varphi_{j,i}}}} = {u_{i} - v_{i}}},} & \left( {{Eq}.\mspace{14mu} 138} \right) \end{matrix}$

where {u_(i), v_(i): i=1, 2, . . . , m} are nonnegative variables. The nonnegative variables {b_(j), c_(j): j=1, 2, . . . , n} and the weights α_(j)=b_(j)−c_(j) may then be defined. The primal problem becomes

$\begin{matrix} {{Minimizing}{\sum\limits_{i = 1}^{m}\left( {u_{i} + v_{i}} \right)}} & \left( {{Eq}.\mspace{14mu} 139} \right) \end{matrix}$

subject to

$\begin{matrix} {{f_{i} = {{\sum\limits_{j = 1}^{n}{\left( {b_{j} - c_{j}} \right)\varphi_{j,i}}} + u_{i} - v_{i}}},{i = 1},2,\ldots \mspace{14mu},m} & \left( {{Eq}.\mspace{14mu} 140} \right) \end{matrix}$

and

b_(j),c_(j),u_(i),v_(i)≧0.  (Eq. 141)

The algorithm may be implemented in two stages: For example, for the first n iterations, the pivotal columns are restricted to b_(j) and c_(j). The vector entering the basis is the one with the largest nonnegative marginal cost, (e.g.,

$\sum\limits_{i = 1}^{m}\varphi_{j,i}$

The vector leaving the basis is chosen among v_(i) and u_(i) for the one causing the maximum reduction in the objective function. Next, the nonbasic u_(i) or v_(i) are interchanged with the basic u_(i) or v_(i). Neither b_(j) or c_(j) is allowed to leave the basis. The criteria for selecting vectors leaving and entering the basis are the same as that in the first stage. Otherwise, the basic vectors b_(j) or c_(j) are interchanged with the corresponding nonbasic vectors c_(j) or b_(j). On making this algorithm efficient, the vector chosen to leave the basis is the one that causes the maximum reduction in the objective function.

On the Multidimensional Neutral Color Conversion Problem, the BR algorithm may be applied to the neutral color conversion by letting z_(i)={(L, a, b)_(i) i=1, 2, . . . , m}. As an example, the color conversion from CIELab to CMY is formulated:

$\begin{matrix} {{{Minimizing}\mspace{14mu} \xi} = {{\sum\limits_{i = 1}^{m}u_{ci}} + v_{ci} + u_{mi} + v_{mi} + u_{yi} + v_{yi}}} & \left( {{Eq}.\mspace{14mu} 142} \right) \end{matrix}$

subject to

$\begin{matrix} {{{\sum\limits_{j = 1}^{n + 4}{\left( {a_{cj} - b_{cj}} \right){F_{cj}\left( z_{i} \right)}}} + u_{ci} + v_{ci}} = C_{i}} & \left( {{Eq}.\mspace{14mu} 143} \right) \\ {{{\sum\limits_{j = 1}^{n + 4}{\left( {a_{mj} - b_{mj}} \right){F_{mj}\left( z_{i} \right)}}} + u_{mi} + v_{mi}} = M_{i}} & \left( {{Eq}.\mspace{14mu} 144} \right) \\ {{{\sum\limits_{j = 1}^{n + 4}{\left( {a_{yj} - b_{yj}} \right){F_{yj}\left( z_{i} \right)}}} + u_{yi} + v_{yi}} = Y_{i}} & \left( {{Eq}.\mspace{14mu} 145} \right) \end{matrix}$

where

a_(cj),b_(cj),a_(mj),b_(mj),a_(yj),b_(yj),u_(i),v_(i)≧0 and

F _(ji)=[1,z(i),φ₁(z _(i)),φ₂(z _(i)), . . . , φ_(n)(z _(i))]_(n+4)

for the C, M, and Y components respectively.

When performing the two-stage BR algorithm, the vectors relating to C, M, and Y are generally be exchanged with the vectors of C, M, and Y respectively. This yields an exemplary condensed initial simplex table for CIELab to CMY conversion as shown below.

One Dimensional Color Manifolds in the CMYK Color Space

One of the more difficult and challenging tasks in color printing is reducing the cost of printing while maintaining quality. Traditionally, the printer color conversion from CMY (or CIELab) to CMYK has been achieved (UCR) and (UCA). As discussed, UCR is the process of removing an equal amount of cyan, magenta, and yellow, and adding the same amount of black. More explicitly, the color components are defined as:

CMY={C,M,Y},CMYK={C,M,Y,K}, and Lab={L,a,b}.

The smallest value of the C, M, Y components is formed as K=min{C, M, Y} and K is used to convert a CMY value to a CMYK value as follows:

C′=C−K,M′=M−K, and Y′=Y−K.

Because of the nonlinearity of toners/inks, this UCR model is in accurate, and introduces unacceptable color conversion errors. Colors resulting from the UCR process are generally dull, muddy, and hue shifted. To mitigate this problem, the UCR process is followed by the UCA correction which consists of adding a relatively small amount of cyan, magenta, and yellow back to the C′, M′, Y′ values respectively to make colors richer and less hue shifted. However, this method is empirical and labor intensive and the conversion error is still quite large.

A solution to this color toner/ink reduction problem consists of approximating a nonlinear continuous and differentiable functions mapping from CIELab to CMY and CMYK to CIELab. To optimize CMY to CMYK conversion, it is assumed that accurate continuous functions describing CMYK to CIELab conversion and CIELab to CMY conversion are as follows:

g(L,a,b)→(C,M,Y)ε(

) and ƒ(C,M,Y)→(L,a,b).

For a CIELab value within the printer gamut, Lab₀=(L₀, a₀, b₀), g(L₀, a₀, b₀)=(C₀,M₀, Y₀). If C₀>0 & M₀>0 & Y₀>0, a (C′₀, M′₀, Y′₀, K′₀) value may be determined such that

$\underset{\underset{X}{}}{f\left( {C_{0}^{\prime},M_{0}^{\prime},Y_{0}^{\prime},K_{0}^{\prime}} \right)} \approx {\underset{\underset{Y}{}}{\left( {L_{0},a_{0},b_{0}} \right)}.}$

Then, the CMYK value x_(n)=(C′_(n),M′_(n),Y′_(n),K′_(n)) may be approximated iteratively such that, for n large enough,

∥ƒ(x _(n))−y _(n)∥≦ε  (Eq. 146)

for some prescribed tolerance ε. Newton's method may be used if an initial point is close to the minimizer algorithm. To determine an initial point for Newton's method a value x₀ is determine such that

err=∥ƒ(x ₀)−y ₀ ∥≦tol,  (Eq. 147)

where tol is an initial acceptable tolerance.

One of the problems in traditional GCR algorithms is that the CIELab value having equal amounts of cyan, magenta, and yellow toner/ink is often not equal to the CIELab value of that same amount of black toner/ink. In other words, the CIELab value of K generally corresponds to unequal amounts of C, M, Y. If the equal amount of C, M, Y, K values are used as the initial point to Newton's algorithm, the above inequality may not be satisfied. A better initial point may be obtained by removing unequal amounts of cyan, magenta, and yellow from the CMY value respectively, and adding the K value. The maximum error using this process may be improved from the typical UCR error value of 14.42 to 10.50.

On this region, for any K, the associated CIELab value is computed using

ƒ(0,0,0,K)→(L _(K) ,a ^(K) ,b ^(K))  (Eq. 148)

Then, the CMY value corresponding to this K is determined using

g(L ^(K) ,a ^(K) ,b ^(K))→(C ^(K) ,M ^(K) ,Y ^(K))  (Eq. 149)

to establish an equivalence

K

(C^(K),M^(K),Y^(K)).  (Eq. 150)

This repeated for a sequence of K values span the range of (0, 1). For any given (C, M, Y) value, there exists a maximum K_(p) value whose corresponding (C^(Kp),M^(Kp),Y^(Kp)) satisfies

C=C_(Kp) or M=M^(Kp) or Y=Y^(Kp)  (Eq. 151)

A sequence of K values is determined such that

K={K_(i)≦K_(p),i=1,2, . . . , p}.  (Eq. 152)

The corresponding (C^(Ki),M^(Ki),Y^(Ki))'s can also be removed from the (C, M, Y) value. Then, the CMY value (C, M, Y) is converted to a sequence of CMYK values such that

CMYK′={(C ^(K′) ^(i) ,M ^(K′) ^(i) ,Y ^(K′) ^(i) ),i=1,2, . . . , p},  (Eq. 153)

where C^(K′) ^(i) =C−C_(K) _(i) , M^(K′) ^(i) =M−M_(K) _(i) , and Y^(K′) ^(i) =Y−Y_(K) _(i) , which yields an optimizing GCR algorithm.

Given an initial CMY, the optimized GCR algorithm may determine an associated (C′,M′,Y′,K′) such that the quality

∥ƒ(C′,M′,Y′,K′)−ƒ(C,M,Y,0)∥  (Eq. 154)

is relatively small. With an initial point close enough to the minimizer, the multidimensional Newton's method may be used to determine the optimal value of (C′,M′, Y′, K′). For example, let

$\begin{matrix} {{f = \begin{pmatrix} f_{L} \\ f_{a} \\ f_{b} \end{pmatrix}},{x = \begin{pmatrix} C \\ M \\ Y \\ K \end{pmatrix}},{y = \begin{pmatrix} L \\ a \\ b \end{pmatrix}},{y = \begin{pmatrix} L_{0} \\ a_{0} \\ b_{0} \end{pmatrix}},} & \left( {{Eq}.\mspace{14mu} 155} \right) \end{matrix}$

where y₀ is the corresponding CIELab value of the original CMY color (e.g., the desired CIELab value for the optimizer of the (C′, M′, Y′, K′)) ƒ(x) then represents the mapping from CMYK to CIELab and expresses as a RBF. For example,

$\begin{matrix} {{{f_{j}(x)} = {{A_{j}x} + a_{j\; 0} + {\sum\limits_{i = 1}^{N}{w_{ji}{\varphi_{j}\left( {{x - c_{i}}} \right)}}}}},} & \left( {{Eq}.\mspace{14mu} 156} \right) \end{matrix}$

where j=1, 2, 3 correspond to {ƒ_(L), ƒ_(a), ƒ_(b)}. Then, ƒ(x+h)≈ƒ(x)+ƒ′(x)h, where ƒ(x) is in the form of Jacobian matrix

$\begin{matrix} {J = {{f^{\prime}(x)} = {\begin{pmatrix} {{\partial f_{L}}/{\partial C}} & {{\partial f_{L}}/{\partial M}} & {{\partial f_{L}}/{\partial Y}} & {{\partial f_{L}}/{\partial K}} \\ {{\partial f_{a}}/{\partial C}} & {{\partial f_{a}}/{\partial M}} & {{\partial f_{a}}/{\partial Y}} & {{\partial f_{a}}/{\partial K}} \\ {{\partial f_{b}}/{\partial C}} & {{\partial f_{b}}/{\partial M}} & {{\partial f_{b}}/{\partial Y}} & {{\partial f_{b}}/{\partial K}} \end{pmatrix}.}}} & \left( {{Eq}.\mspace{14mu} 157} \right) \end{matrix}$

For each iteration k, x_(k+1)=x_(k)−J(x_(k))^(−I)(ƒ(y_(k))−ƒ(y₀)). This iteration stops when ∥(f(x_(n))−y₀)∥≦10⁻¹².

The dimension of the Jacobian matrix is generally 3×4. In order to change it to an invertible Jacobian matrix. The black value of the (C,M, Y,K) color remains unchanged. The Jacobian matrix in Equation 149 is then changed to

$\begin{matrix} {J = {{f^{\prime}(x)} = \begin{pmatrix} {{\partial f_{L}}/{\partial C}} & {{\partial f_{L}}/{\partial M}} & {{\partial f_{L}}/{\partial Y}} \\ {{\partial f_{a}}/{\partial C}} & {{\partial f_{a}}/{\partial M}} & {{\partial f_{a}}/{\partial Y}} \\ {{\partial f_{b}}/{\partial C}} & {{\partial f_{b}}/{\partial M}} & {{\partial f_{b}}/{\partial Y}} \end{pmatrix}}} & \left( {{Eq}.\mspace{14mu} 158} \right) \end{matrix}$

Solving the Equation 149 using SVD obtains

$\begin{matrix} {x_{k + 1} = {\begin{pmatrix} C_{k + 1} \\ M_{k + 1} \\ Y_{k + 1} \\ K \end{pmatrix}.}} & \left( {{Eq}.\mspace{14mu} 159} \right) \end{matrix}$

At each iteration, check if the C, M, Y values in the converted (C, M, Y K) analyzer to determine whether they are in the range of [0, I], (e.g., if x_(k+1) ^(i)≧1 or x_(k+1) ^(i)≦0, where i=1, 2, 3 correspond to C, M and Y component respectively). If the value of the i^(th) component is out of the range,

$\begin{matrix} {x_{k + 1}^{i} = \left\{ \begin{matrix} {1,{{x(i)} \geq 1}} \\ {0,{{x(i)} \leq 0.}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 160} \right) \end{matrix}$

The i^(th) component remains unchanged for the rest of the iterations. The column that corresponds to this component from the Jacobian matrix J is then removed, and the constraint on the black component is relaxed by adding the partial derivatives with respect to black in the matrix J. For example, if x(2)≧1, the magenta component is removed. The Jacobian matrix J_(k+1) becomes

$\begin{matrix} {{J_{k + 1} = \begin{pmatrix} {{\partial f_{L}}/{\partial C}} & {{\partial f_{L}}/{\partial M}} & {{\partial f_{L}}/{\partial Y}} \\ {{\partial f_{a}}/{\partial C}} & {{\partial f_{a}}/{\partial M}} & {{\partial f_{a}}/{\partial Y}} \\ {{\partial f_{b}}/{\partial C}} & {{\partial f_{b}}/{\partial M}} & {{\partial f_{b}}/{\partial Y}} \end{pmatrix}}{and}} & \left( {{Eq}.\mspace{14mu} 161} \right) \\ {x_{k + 1} = {\begin{pmatrix} C_{k + 1} \\ 1.0 \\ Y_{k + 1} \\ K_{k + 1} \end{pmatrix}.}} & \left( {{Eq}.\mspace{14mu} 162} \right) \end{matrix}$

Generally, any given CMY value of (C>0 & M>0 & Y>0) converges to a CMYK value with the CIELab value difference practically zero.

As described above, each black level K, (e.g., CMYK=(0, 0, 0, K) can be mapped to a CMY value of (C_(k),M_(k), Y_(k)). For any CMY value (C>0 & M>0 & y>0), there exists a set of values {K_(i), i=1, 2, . . . , p} in which a corresponding {C^(ki), M^(ki), Y^(ki), i=1, 2, . . . , p} may be removed from C, M, and Y, (e.g., C≧C^(ki) & M≧M^(ki) & Y≧Y^(ki), i=1, 2, . . . , p). Generally, for one CMY value, there exists a set of CMYK values {C−C^(ki), M−M^(ki), Y−Y^(ki), K_(i), i=1, 2, . . . , p} that can be used for the initial points to the multidimensional Newton's optimization algorithm. Thereafter, the set of values ƒ⁻¹(L₀, a₀, b₀) may be determined as a one-dimensional manifold based on the pre-image theorem.

For example, X→Y be a smooth map between manifolds. Thus, a point is a regular value off if for all x: xεƒ⁻¹(y) and the map dƒ: T_(x)X→T_(y)Y is subjective. Here, T_(x)X and T_(y)Y are the tangent spaces of X and Y at the points x and y. Again, with ƒ: X→Y being a smooth map, yεY may be a regular value of ƒ. Accordingly, x: xεƒ⁻¹(y) is a submanifold of X with dim ƒ⁻¹(y)=dimX−dimY.

For color conversion, A={set of elements of CMY K} and B={set of elements of CIELab}. A function ƒ=(ƒ_(L), ƒ_(a), ƒ_(b)) that maps A to B is in the form of RBFs described in Eq. 156 and ƒ is continuous and differentiable. For a CIELab value yεB, if the Jacobian matrix described in Eq. 157 at y is full rank, (e.g., rank is equal to 3), then y is a regular point of f. Generally, there exists a manifold S=ƒ⁻¹(y), S⊂A with the dimension of one.

These one-dimensional manifolds based on the pre-image theorem are then determined. For example, for any CIELab value y that is inside the printer gamut, obtain the CMY value using g(y)→(C, M, Y). Then, determine whether min(C, M, y)=0. If soy is not a regular point since its Jacobian matrix will be rank deficient. Otherwise, if min(C, M, Y>0, the initializing algorithm is performed to obtain the sequence of the CMYK value CMYK′ described in Eq.155.

The Jacobian matrix is monitored at each starting point to decide if y is a regular point. If y is a regular point, the optimized GCR algorithm is performed to obtain optimized value for each starting point, (e.g., obtain the 1-dimensional manifold S⊂A, and S=ƒ⁻¹(y)). When there exists a level set of CMYK values S=ƒ⁻¹(y) that correspond to the same CIELab value, yεB, the optimal solution for the toner/ink selection can be obtained for a given objective function.

The objective function can be toner/ink cost based. That is, the objective function is to minimize the cost of printing or toner/ink coverage. For example, min(C+M+Y+K) is the minimum amount of C+M+Y+K obtained from the level set. The data in FIG. 5 for the original CMY value {0.7, 1, 1} results in the total amount of toner/ink being 270%. The minimum total amount of toners/inks in the CMYK level set is 84.2% corresponding the CMYK value of {0.066186, 0.031126, 0, 0.74461} resulting in a total toner savings of 185.8%. Because values in a CMYK level set correspond to the same CIELab value, the toner savings is achieved with little or no quality degradation.

Color Conversion with Ink Limitations

Another challenging task in color printing relates to the limitation of the amount of toner/ink coverage on a page to avoid excessive bleed-through. This task is particularly important for inkjet printing. When there exists a limit for the maximum amount of toner/ink coverage allowed per spot color printing, color conversion in highly saturated color regions or dark neutral color regions are generally affected. The color conversion affected by this limit includes the CIELab to CMY color conversion. The L₁ norm approximation may provide a suitable solution for solving the neutral color conversion problem and the L_(∞) approximation may be suitable for color conversion in saturated color regions. For both approaches, RBFs with linear programming may be employed. The existence of a toner/ink limitation may be added as a constraint in the linear programming problem formulations. For example, the color conversion with the toner/ink limitation problem may now be formulated via both L₁ and L_(∞) approximation algorithms. The L₁ algorithm may be a modified BR primal algorithm with an added constraint while the L_(∞) algorithm may be developed based on the BP dual algorithm to extend the three-stage algorithm to a four-stage algorithm.

To analyze ink limitation problem using the L₁ approximation, g(z) is used to represent the mapping from CIELab to CMY in the form of an RBF as:

$\begin{matrix} {{{g_{k}\left( z_{j} \right)} = {{A_{k}z_{j}} + \alpha_{k\; 0} + {\sum\limits_{i = 1}^{n}{\alpha_{ki}{\varphi_{k}\left( {{z_{j} - c_{i}}} \right)}}}}},} & \left( {{Eq}.\mspace{14mu} 163} \right) \end{matrix}$

where k=1, 2, 3 corresponding to (g_(C), g_(M), g_(Y)). Then, z_(j)={L, a, b}_(j), where i=1, 2, . . . , n and j=1, 2, . . . , m. The dimension of A_(k) is 1×3 corresponding to the dimension of the domain. The dimension of a₀ is 1. Therefore, there are n+4 coefficients in such that Eq. 163 may be written as:

$\begin{matrix} {{g_{k}\left( z_{j} \right)} = {\underset{F_{k}{(z_{j})}}{\underset{}{\begin{pmatrix} {z_{j},1,{\varphi_{k}\left( {{z_{j} - c_{1}}} \right)},} \\ {{\varphi_{k}\left( {{z_{j} - c_{2}}} \right)},...\mspace{11mu},} \\ {\varphi_{k}\left( {{z_{j} - c_{i}}} \right)} \end{pmatrix}}}\underset{\underset{w_{k}}{}}{\begin{matrix} \begin{pmatrix} A_{k}^{T} \\ a_{k\; 0} \\ a_{k\; 1} \\ \alpha_{k\; 2} \\ \ldots \\ \alpha_{kn} \end{pmatrix} \\ {n + 4} \end{matrix}}}} & \left( {{Eq}.\mspace{14mu} 164} \right) \\ {\mspace{59mu} {= {\sum\limits_{i = 1}^{n + 4}{w_{ki}{{F_{ki}\left( z_{i} \right)}.}}}}} & \left( {{Eq}.\mspace{14mu} 165} \right) \end{matrix}$

The L₁ error for each data point of component of C, M and Y using nonnegative variables are defined as

$\begin{matrix} {{{g_{k}\left( z_{i} \right)} - {\sum\limits_{i = 1}^{n + 4}{w_{ki}{F_{ki}\left( z_{i} \right)}}}} = {u_{kj} - v_{kj}}} & \left( {{Eq}.\mspace{14mu} 166} \right) \end{matrix}$

and the coefficients for each component using nonnegative variables are defined as w_(ki)=b_(ki)−c_(ki), where u_(kj), v_(kj), b_(ki), c_(ki)≧0. The primal linear programming problem in the L₁ sense may be illustrated by the constraint of the toner/ink limit applying to the color data with a toner/ink summation exceeding the total amount of toner or ink (e.g., C+M+Y+K>L, where L is the total amount of toner or ink). In this regard,

$\begin{matrix} {{{Minimize}\mspace{14mu} \xi} = {{\sum\limits_{j = 1}^{m}u_{1\; j}} + v_{1\; j} + u_{2\; j} + v_{2\; j} + u_{3\; j} + v_{3\; j}}} & \left( {{Eq}.\mspace{14mu} 167} \right) \end{matrix}$

subject to

$\begin{matrix} {\mspace{79mu} {{{\sum\limits_{i = 1}^{n + 4}{\left( {a_{1\; j} - b_{1\; j}} \right){F_{1\; j}\left( z_{j} \right)}}} + u_{1\; j} + v_{1\; j}} = C_{j}}} & \left( {{Eq}.\mspace{14mu} 168} \right) \\ {\mspace{79mu} {{{\sum\limits_{i = 1}^{n + 4}{\left( {a_{2\; j} - b_{2\; j}} \right){F_{2\; j}\left( z_{j} \right)}}} + u_{2\; j} + v_{2\; j}} = M_{j}}} & \left( {{Eq}.\mspace{14mu} 169} \right) \\ {\mspace{79mu} {{{\sum\limits_{i = 1}^{n + 4}{\left( {a_{3\; j} - b_{3\; j}} \right){F_{3\; j}\left( z_{j} \right)}}} + u_{3\; j} + v_{3\; j}} = Y_{j}}} & \left( {{Eq}.\mspace{14mu} 170} \right) \\ {{{{{\sum\limits_{i = 1}^{n + 4}{\left( {a_{1\; j} - b_{1\; j}} \right)F_{1\; j}\left( z_{jl} \right)}} + {\sum\limits_{i = 1}^{n + 4}{\left( {a_{2\; j} - b_{2\; j}} \right)F_{2\; j}\left( z_{jl} \right)}} + {\sum\limits_{i = 1}^{n + 4}{\left( {a_{3\; j} - b_{3\; j}} \right){F_{3\; j}\left( z_{jl} \right)}}}} \leq L},\mspace{79mu} {where}}\mspace{11mu} \; {{J = {\left\{ j_{l} \middle| {{C_{jl} + M_{jl} + Y_{jl}} > L} \right\} \subseteq \left\{ {{i = 1},2,{\ldots \mspace{11mu} m}} \right\}}},u_{kj},v_{kj},b_{ki},{c_{ki} \geq 0},{k = 1},2,3,{i = 1},2,\ldots \mspace{11mu},n,{{{and}\mspace{14mu} j} = 1},2,\ldots \mspace{11mu},{m.}}} & \left( {{Eq}.\mspace{14mu} 171} \right) \end{matrix}$

With this in mind, a condensed initial simplex table for the CIELab to CMY conversion with the toner/ink limitation L is established in the table below where p denotes the number of color data points that exceed the toner/ink limit. The slack variables s_(i)'s are then added in Eq. 171.

The slack variables s_(i)'s are then added in Eq. 171.

The modified BR's algorithm may then be described in two stages. In the first stage, there are three sets of n iterations for the C, M, and Y components respectively. In each of the n iterations, the pivotal columns are restricted to b_(i) and c_(i). The vector entering the basis is the one with the largest nonnegative marginal cost. The vector leaving the basis is chosen from v_(j) and u_(j) for the one causing the maximum reduction in the objective function. The interchange of vectors leaving and entering the basis are these performed within the same color component.

In the next stage, the nonbasic u_(j) or v_(j) are interchanged with the basic u_(j), v_(j), or s_(jl). Neither b_(i) or c_(i) is allowed to leave the basis. The criteria for selecting vectors leaving and entering the basis are the same as that in the first stage. The interchange of vectors v_(j) and u_(j) leaving and entering the basis may be performed within the same color component. However, w_(jl) can be interchanged with the v_(j) and u_(j) vector for any color component. Once the vector s_(jl) leaves the basis, it is generally not allowed to enter back onto the basis. The iteration stops when all the marginal costs are non-positive. Thereafter, basic vectors b_(i) or c_(i) may be interchanged with the corresponding nonbasic vectors c_(j) or b_(j).

As discussed above, the BP dual algorithm may be a relatively good algorithm for color conversion using L_(∞) approximation. For the color conversion with the toner/ink limitation, a constraint may be added in the problem formulation of the BP dual algorithm using the L_(∞) approximation.

In this regard, the L_(∞) errors may be defined for component of C, M and Y:

$\begin{matrix} {\xi_{c\;} = {\max\limits_{1 \leq j \leq m}{{{\sum\limits_{i = 1}^{n + 4}{w_{ci}{F_{ci}\left( z_{j} \right)}}} - C_{j}}}_{\infty}}} & \left( {{Eq}.\mspace{14mu} 172} \right) \\ {\xi_{m\;} = {\max\limits_{1 \leq j \leq m}{{{\sum\limits_{i = 1}^{n + 4}{w_{mi}{F_{mi}\left( z_{j} \right)}}} - M_{j}}}_{\infty}}} & \left( {{Eq}.\mspace{14mu} 173} \right) \\ {\xi_{y\;} = {\max\limits_{1 \leq j \leq m}{{{\sum\limits_{i = 1}^{n + 4}{w_{yi}{F_{yi}\left( z_{j} \right)}}} - Y_{j}}}_{\infty}}} & \left( {{Eq}.\mspace{14mu} 174} \right) \end{matrix}$

Then, the problem may be formulated in the primal form by minimizing ξ=ξ_(c)+ξ_(m)+ξ_(y), subject to

$\begin{matrix} {{\xi_{c} + {\sum\limits_{i = 1}^{n + 4}{\alpha_{ci}{F_{ci}\left( z_{j} \right)}}}} \geq C_{j}} & \left( {{Eq}.\mspace{14mu} 175} \right) \\ {{\xi_{c} - {\sum\limits_{i = 1}^{n + 4}{\alpha_{ci}{F_{ci}\left( z_{j} \right)}}}} \geq {- C_{j}}} & \left( {{Eq}.\mspace{14mu} 176} \right) \\ {{\xi_{m} + {\sum\limits_{i = 1}^{n + 4}{\alpha_{mi}{F_{mi}\left( z_{j} \right)}}}} \geq M_{j}} & \left( {{Eq}.\mspace{14mu} 177} \right) \\ {{\xi_{m} - {\sum\limits_{i = 1}^{n + 4}{\alpha_{mi}{F_{mi}\left( z_{j} \right)}}}} \geq {- M_{j}}} & \left( {{Eq}.\mspace{14mu} 178} \right) \\ {{\xi_{y} + {\sum\limits_{i = 1}^{n + 4}{\alpha_{yi}{F_{yi}\left( z_{j} \right)}}}} \geq Y_{j}} & \left( {{Eq}.\mspace{14mu} 179} \right) \\ {{\xi_{y} - {\sum\limits_{i = 1}^{n + 4}{\alpha_{yi}{F_{yi}\left( z_{j} \right)}}}} \geq {- Y_{j}}} & \left( {{Eq}.\mspace{14mu} 180} \right) \\ {{{- {\sum\limits_{i = 1}^{n + 4}\left( {{\alpha_{ci}{F_{ci}\left( z_{jl} \right)}} + {\alpha_{mi}{F_{mi}\left( z_{jl} \right)}} + {\alpha_{yi}{F_{yi}\left( z_{jl} \right)}}} \right)}} \leq {- L}},} & \left( {{Eq}.\mspace{14mu} 181} \right) \end{matrix}$

where ξ_(c), ξ_(m), ξ_(y)≧0, α₁, α₂, . . . α_(n+4) unrestricted, z_(j)={L, a, b}_(j), j=1, 2, . . . , m, J={jl|C_(jl)+M_(jl)+Y_(jl)>L}⊂{j=1, 2, . . . , m}, and F_(j)=[1, z_(j), φ₁(z_(j)), φ₂(z_(j)), . . . , φ_(n)(z_(j))]_(n+4),

Transformed from the primal formulation, the dual formulation is performed by maximizing

$\begin{matrix} {{{\sum\limits_{i = 1}^{m}\left\lbrack {{\left( {{\underset{\_}{\sigma}}_{ci} - {\underset{\_}{\tau}}_{ci}} \right)C_{i}} + {\left( {{\underset{\_}{\sigma}}_{mi} - {\underset{\_}{\tau}}_{mi}} \right)M_{i}} + {\left( {{\underset{\_}{\sigma}}_{yi} - {\underset{\_}{\tau}}_{yi}} \right)Y_{i}}} \right\rbrack} - {L{\sum\limits_{i_{k} = 1}^{p}{\underset{\_}{\rho}}_{i_{k}}}}},} & \left( {{Eq}.\mspace{14mu} 182} \right) \end{matrix}$

subject to

$\begin{matrix} {\sum\limits_{j = 1}^{m}\left\lbrack {{{\left( {{\underset{\_}{\sigma}}_{cj} - {\underset{\_}{\tau}}_{cj}} \right)F_{c_{ij}}^{T}} - {\sum\limits_{{jl} = 1}^{p}{{\underset{\_}{\rho}}_{j}F_{c_{{ij}_{l}}\;}^{T}}}} = 0} \right.} & \left( {{Eq}.\mspace{14mu} 183} \right) \\ {\sum\limits_{j = 1}^{m}\left\lbrack {\left( {{\underset{\_}{\sigma}}_{cj} - {\underset{\_}{\tau}}_{cj}} \right) \leq 1} \right.} & \left( {{Eq}.\mspace{14mu} 184} \right) \\ {\sum\limits_{j = 1}^{m}\left\lbrack {{{\left( {{\underset{\_}{\sigma}}_{mj} - {\underset{\_}{\tau}}_{mj}} \right)F_{m_{ij}}^{T}} - {\sum\limits_{{jl} = 1}^{p}{{\underset{\_}{\rho}}_{j}F_{m_{{ij}_{l}}\;}^{T}}}} = 0} \right.} & \left( {{Eq}.\mspace{14mu} 185} \right) \\ {\sum\limits_{j = 1}^{m}\left\lbrack {\left( {{\underset{\_}{\sigma}}_{mj} - {\underset{\_}{\tau}}_{mj}} \right) \leq 1} \right.} & \left( {{Eq}.\mspace{14mu} 186} \right) \\ {\sum\limits_{j = 1}^{m}\left\lbrack {{{{\left( {{\underset{\_}{\sigma}}_{yj} - {\underset{\_}{\tau}}_{yj}} \right)F_{y_{ij}}^{T}} - {\sum\limits_{{jl} = 1}^{p}{{\underset{\_}{\rho}}_{j}F_{y_{{ij}_{l}}\;}^{T}}}} = 0},} \right.} & \left( {{Eq}.\mspace{14mu} 187} \right) \\ {\sum\limits_{j = 1}^{m}\left\lbrack {{\left( {{\underset{\_}{\sigma}}_{yj} - {\underset{\_}{\tau}}_{yj}} \right) \leq 1},} \right.} & \left( {{Eq}.\mspace{14mu} 188} \right) \end{matrix}$

and

σ _(j),τ _(j),ρ _(j)≧0,j=1,2, . . . , m,  (Eq. 189)

where z_(j)={L, a, b}_(j), j=1, 2, . . . , m, J={jl|C_(jl)+M_(jl)+Y_(jl)>L}⊂{j=1, 2, . . . , m}, F_(j)=[1, z_(j), φ₁(z_(j)), φ₂(z_(j)), . . . , φ_(n)(z_(j))]_(n+4), F_(jl)=[1, z_(jl), φ_(jl)(z_(jl)), φ_(j2)(z_(jl)), . . . , φ_(n)(z_(jl))]_(n+4), φ_(ij)=φ_(i)(z_(j)), and p denotes the number of color data points that exceed the toner/ink limit.

A modified BP L_(∞) approximation algorithm may be used for solving the color conversion from CIELab to CMY with a toner/ink limitation L based on the condensed initial simplex table below.

In the first stage of the BP algorithm, the first n simplex iterations are performed. For example, the α_(c) vectors may be moved out of the basis. The vector entering the basis is selected from s_(c) to correspond to that with the largest absolute reduced cost. The am vectors may be moved out of the basis. The vector entering the basis is selected from s_(m) to correspond to that with the largest absolute reduced cost. The α_(y) vectors may be moved out of the basis. The vector entering the basis is selected from s_(y) to correspond to that with the largest absolute reduced cost.

In the second stage of the BP algorithm, ξ_(c) is forced out of the basis. The vector entering the basis is selected from the remaining nonbasic vectors s_(c) and α_(c) is not allowed back onto the basis. ξ_(m) is forced out of the basis. The vector entering the basis is selected from the remaining nonbasic vector s_(m) and α_(m) is not allowed to move back onto the basis. ξ_(y) is forced out of the basis. The vector entering the basis is selected from the remaining nonbasic vector s_(y) and α_(y) is not allowed to move back onto the basis.

In the third stage of the BP algorithm, the pivotal column is chosen corresponding to the most negative marginal cost. The pivotal row may then be chosen by the ratio selection rule. The interchange of nonbasic vectors s_(i) and basic vectors s_(j) may be performed within the same color component. The iteration continues until all marginal costs corresponding to the nonbasic vectors s_(c), s_(m), and s_(y), are nonnegative. The vector s_(cmy) is not allowed to enter the basis.

A fourth stage is added to perform simplex iteration interchanging nonbasic vectors including all the remaining vectors s_(c), s_(m), s_(y), and s_(cmy) and the basic vectors in the basis. Nonbasic vectors s_(c), s_(m), and s_(y) interchange with basic vectors of s_(c), s_(m), and s_(y) respectively. The nonbasic vectors s_(c), s_(m), and s_(y) interchange with any basic vectors of s_(c), s_(m), and s_(y) ending when the marginal costs are nonnegative. The function φ thus satisfies the Haar condition as described in above.

Gamut Mapping

The inventive embodiments of out-of-gamut conversion that include CIELab to CMYK gamut conversion may be described as follows. First, the thresholds for color difference in hue angle Th and color difference in lightness TL based on the perceptual color difference tolerance are defined. Next, the output device gamut boundary gmt0 specified in the CIELab color space is defined. This boundary may be expanded into a few layers with a small increment each time (e.g., {gmt_(i), i=1, 2, N}). In each layer, the CIELab values are {(L, a, b)_(ij), j=1, 2, . . . , m_(i)}. The corresponding color in CIELch is {(L,c,h)_(ij), j=1, 2, . . . , m_(i)}, where mi is the number of the CIELab colors in the layer i. Then the CMY value is obtained using

g ₀((L,a,b)_(1j))→(C,M,Y)_(1j),  (Eq. 190)

where g0 is the mapping function from CIELab to CMY based on the inside gamut colors. The CIELab data in the first expanded layer is (L, a, b)₁={(L, a, b)_(1j), j=1, 2, . . . , m₁}. The number of color data points in gmt1 is (C,M,Y)₁={(C,M,Y)_(1j), j=1, 2, . . . , m₁}, m₁. If any C, M, and Y component value in (C, M, Y) 1 is smaller than 0 or greater than 1, then the value is changed to 0 or 1 respectively. The (C, M, Y)1 may then be changed to (C, M, Y)′1. (L, a, b)′1 is obtained using

ƒ₀(C′ _(1j) ,M′ _(1j) ,Y′ _(1j),0)→(L′ _(1j) ,a′ _(1j) ,b′ _(1j))  (Eq 191)

where ƒ₀ is the mapping function from CMYK to CIELab based on the inside gamut colors. Then (L, a, b)′ 1 to (L, C, h)′ 1 is converted and the lightness difference ({dL_(1j), j=1, 2, . . . , m₁} and the hue difference ({dh_(1j), j=1, 2, . . . , m₁} are calculated for each data point between (L, C, h)₁ and (L, C, h)′₁. If dh_(1j)>T_(h) or dL1j>T_(L), (L, C, h)′₁ is converted to (L, C, h)″₁ by making the following changes in L′_(1j) and h′_(1j):

$\begin{matrix} {L = \left\{ \begin{matrix} {{L_{1\; i} + T_{L}},{L_{1\; j}^{\prime} > L_{1\; j}}} \\ {{L_{1\; i} + T_{L}},{L_{1\; j}^{\prime} < L_{1\; j}}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 192} \right) \\ {h = \left\{ \begin{matrix} {{h_{1\; j} + T_{h}},{h_{1\; j}^{\prime} > h_{1\; j}}} \\ {{h_{1\; j} + T_{h}},{h_{1\; j}^{\prime} < {h_{1\; j}.}}} \end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 193} \right) \end{matrix}$

The steps may be repeated until the (C, M, Y)₁ values are found to meet the criteria of T_(h) and T_(L). Then all out-of-gamut CIELab values (L, a, b)₁ may be mapped in gmt1 with (C, M, Y)₁. Models of g₁ and f₁ are then generated using the data points in gmt1:

g ₁((L,a,b))→(C,M,Y) and  (Eq 194)

ƒ₁((C,M,Y)1)→(L,a,b).  (Eq. 195)

Each of the above steps may then be repeated and the CIELab data interpolated in {gmt_(i), i=2, 3, . . . , N} to find {(C,M,Y)_(ij), j=1, 2, . . . , m_(i)}. The iteration stops when the last color layer reaches optimized color ranges (e.g., as described in U.S. Pat. No. 6,719,392, the entire contents of which are incorporated by reference)

When the RBF models f_(i) and g_(i) are generated based on the data points in the gamut layer gmt_(i), the data used in generating these models may contain two parts. The first part includes the data that fits the model g_((i−1)) and the data of f_((i−1)) that is generated based on the gamut layer i−1. The second part includes the data not fitting g_((i−1)) and f_((i−1)). This second part has C, M, or Y values either below 0 or above 1 using f_((i−1)) and is clipped to either 0 and 1 and refined with the thresholds T_(h) and T_(L).

In doing so, the models generated for each gamut layer are continuous and smooth. To summarize, the gamut mapping direction is obtained by creating the numerical model guided by a perceptual color difference model. The advantages of this technique are that more levels are obtained from this gamut mapping algorithm. Moreover, out-of-gamut colors in a close neighborhood vary continuously. The direction of the gamut mapping is not fixed, rather it varies smoothly to provide aesthetically pleasing color mappings.

FIG. 6 is a flowchart illustrating a method 600 of converting color values of image data to a color space in one exemplary embodiment of the invention. In this embodiment, the color region identifier 101 first receives image data that includes color values in the process element 601 such that the image data may be converted to a desired color space. The color region identifier 101 then selects clusters of the color values in the process element 602 such that color regions may be identified. In doing so, the color region identifier 101 determines centers of the clusters in the process element 603 and determines boundaries of the clusters to define the color regions in the process element 604. For example, the color region identifier 101 may select Voronoi sets of the color values, determine random numerical centers of those sets and iteratively determine the boundaries of the clusters based on the centers using an OLS algorithm.

From there, the optimization module 102 may process subsets of the color values in the process element 605. For example, the optimization module 102 may optimize the number of the color regions through a covariance matrix-based processing of each color region. In doing so, the optimization module 102 may determine a rank of each covariance matrix and, based on that rank, determine whether certain color regions should be removed from subsequent processing. That is, the optimization module 102 determines whether the cluster accurately defines the color region in the process element 606. If the processed cluster does not accurately define a color region, the cluster of color values is removed from subsequent processing in the process element 608. The color values thereof may then be reallocated to adjacent clusters in the process element 609 such that they may be processed in the remaining color regions in the process element 605. If a cluster does accurately define a color region, the color region is then prepared for color conversion in the process element 607. For example, once all remaining clusters have a full rank covariance matrix, the image data may be prepared for color conversion via the RBF processing of the converter 103 to convert the image data to a desired color space.

Although specific embodiments were described herein, the scope of the invention is not limited to those specific embodiments. The scope of the invention is intended to be defined by the following claims and any equivalents thereof. For example, the invention is not intended to be limited to any particular type of printer. The clustering and color conversion techniques described herein may be employed in either ink or toner based printers. Moreover, the clustering and color conversion techniques may be employed in printers ranging in size from small household printers to large commercial printers. In one embodiment, the clustering and color conversion techniques may be implemented as software instructions operable with a host system to convert colors as needed. In this regard, the software instructions may direct a processing system, such as a general purpose computer, to cluster color values into color regions for RBF based color conversion prior to printing or display with another imaging device (e.g., a computer monitor). 

1. A color conversion system operable to identify color regions in image data and convert the image data to a color space, the system comprising: a color region identifier operable to select color values in the image data, determine numerical centers of the selected color values, and generate color regions based on the selected color values and the numerical centers; and an optimization module operable to refine the numerical center of each color region, determine a boundary for each color region based on the refined center, and remove a portion of the numerical centers to optimize a number of the color regions for color conversion processing.
 2. The color conversion system of claim 1, wherein the optimization module is further operable to compute a covariance matrix of a color region to optimize the number of the color regions.
 3. The color conversion system of claim 2, wherein the optimization module is further operable to remove the color region from subsequent processing based on a rank of the covariance matrix of the color region.
 4. The color conversion system of claim 3, wherein the optimization module is further operable to configure the color values of the removed color region with an adjacent color region.
 5. The color conversion system of claim 1, wherein the color region identifier is further operable to randomly select the numerical center of the selected color values when determining the numerical center of the selected color values.
 6. The color conversion system of claim 1, wherein the optimization module is further operable to determine a Voronoi set of the selected color values, iteratively update the Voronoi set, and refine the numerical center of the selected color values via an orthogonal least squares algorithm.
 7. The color conversion system of claim 1, further comprising a converter operable to generate a numerical model of the color region using a radial basis function and convert the color region to a color space of an imaging device.
 8. The color conversion system of claim 1, wherein the optimization module is further operable to determine a boundary for each color region based on human perceptions properties.
 9. A method of determining color regions in image data, the method comprising: selecting clusters of color values in the image data; determining random center values of the clusters; iteratively determining boundaries of the clusters to define the color regions; and processing subset color values of the clusters to remove clusters from subsequent processing.
 10. The method of claim 9, wherein processing subset color values comprises: generating a covariance matrix of each cluster; determining a rank of each covariance matrix; and analyzing the rank of each covariance matrix to determine subsequent processing of a respective cluster and optimize a number of the clusters.
 11. The method of claim 10, further comprising: moving color values of a removed cluster to an adjacent cluster in response to analyzing the rank of the covariance matrix of the removed cluster; and repeating said processing subset color values of the clusters until each cluster includes a full rank covariance matrix.
 12. The method of claim 9, wherein determining the random center values of the clusters comprises: generating a Voronoi set of the selected color values; iteratively updating the Voronoi set; and refining the center values of the clusters via an orthogonal least squares algorithm.
 13. The method of claim 9, further comprising: generating numerical models of the color regions using radial basis functions; and converting the color regions to a color space based on the numerical models.
 14. The method of claim 9, wherein iteratively determining boundaries of the clusters to define the color regions is based at least in part on human perceptions properties.
 15. A printing system, comprising: a printer operable to print to a tangible medium; and a printer controller operable to process a print job from a host system and transfer the processed print job to the printer, wherein the print job includes image data having color values and wherein the printer controller comprises: a color region identifier operable to select the color values in the image data, determine numerical centers of the selected color values, and generate color regions based on the selected color values and the numerical centers; an optimization module operable to refine the center of the color region and determine a boundary for each color region based on the refined center and remove a portion of the numerical centers to optimize a number of the color regions for color conversion processing; and a converter operable to convert the color regions to a color space prior to printing.
 16. A color conversion system operable to identify color regions in image data and convert the image data to a color space, the system comprising: a color region identifier operable to select clusters of color values in the image data and determine random center values of the clusters; and an optimization module operable to iteratively determine boundaries of the clusters to define the color regions and process subset color values of the clusters to remove clusters from subsequent processing.
 17. The color conversion system of claim 16, wherein the optimization module is further operable to generate a covariance matrix of each cluster, determine a rank of each covariance matrix, and analyze the rank of each covariance matrix to determine subsequent processing of a respective cluster and optimize a number of the clusters.
 18. The color conversion system of claim 17, wherein the optimization module is further operable to move color values of a removed cluster to an adjacent cluster in response to analyzing the rank of the covariance matrix of the removed cluster and process subset color values of the clusters until each cluster includes a full rank covariance matrix.
 19. The color conversion system of claim 16, wherein the color region identifier is further operable to generate a Voronoi set of the selected color values, iteratively update the Voronoi set, and refine the center values of the clusters via an orthogonal least squares algorithm.
 20. The color conversion system of claim 16, further comprising a converter operable to generate a numerical model of the color regions using radial basis functions and to convert the color regions to a color space of an imaging device based on the numerical models. 